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ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannot[ESslashwidehatESleftbrackESslashxESrightbrack, ESslashwidehatESleftbrackESslashsESrightbrack] &=ESslasharg ESslashmax_ESleftbrackESslashx,ESslashsESrightbrack p(ESslashx,ESslashs|ESslashy) ESslashnonumber ESslashESslash &= ESslasharg ESslashmin_ESleftbrackESslashx,ESslashsESrightbrack - ESslashlog p(ESslashy|ESslashx) - ESslashlog p(ESslashx|ESslashs) - ESslashlog p(ESslashs) ESslashnonumber ESslashESslash &= ESslasharg ESslashmin_ESleftbrackESslashx,ESslashsESrightbrack ESslashLy(ESslashx) - ESslashlog p(ESslashx|ESslashs) + R(ESslashsss) ESannotend) /EscapedSubject (ESannoes%5B%5Cwidehat%7B%5Cx%7D,%20%5Cwidehat%7B%5Cs%7D%5D%20&=%5Carg%20%5Cmax_%7B%5Cx,%5Cs%7D%20p(%5Cx,%5Cs%7C%5Cy)%20%5Cnonumber%20%5C%5C%0A&=%20%5Carg%20%5Cmin_%7B%5Cx,%5Cs%7D%20-%20%5Clog%20p(%5Cy%7C%5Cx)%20-%20%5Clog%20p(%5Cx%7C%5Cs)%20-%20%5Clog%20p(%5Cs)%20%5Cnonumber%20%5C%5C%0A&=%20%5Carg%20%5Cmin_%7B%5Cx,%5Cs%7D%20%5CLy(%5Cx)%20%20-%20%5Clog%20p(%5Cx%7C%5Cs)%20+%20R(%5Csss)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashLy(ESslashx)= - ESslashlog p(ESslashy | ESslashx) ESannotend) /EscapedSubject (ESannoes%5CLy(%5Cx)=%20-%20%5Clog%20p(%5Cy%20%7C%20%5Cx)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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%ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmin_ESleftbrackESslashx ESslashin ESslashR^NESrightbrack ESslash| ESslashy - ESslashA ESslashx ESslash|_2^2 + ESslashlambda ESslash| ESslashx ESslash|_0ESannotend) /EscapedSubject (ESannoes%5Cmin_%7B%5Cx%20%5Cin%20%5CR%5EN%7D%20%5C%7C%20%5Cy%20-%20%5CA%20%5Cx%20%5C%7C_2%5E2%20+%20%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_0ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannot[ESslashwidehatESleftbrackESslashxESrightbrack, ESslashwidehatESleftbrackESslashsESrightbrack] &= ESslasharg ESslashmin_ESleftbrackESslashx,ESslashsESrightbrack ESslashLy(ESslashx) - ESslashsum_ESleftbracki=1ESrightbrack^N ESslashlog p(x_i | s_i) + R(ESslashsss) ESannotend) /EscapedSubject (ESannoes%5B%5Cwidehat%7B%5Cx%7D,%20%5Cwidehat%7B%5Cs%7D%5D%20%0A&=%20%5Carg%20%5Cmin_%7B%5Cx,%5Cs%7D%20%5CLy(%5Cx)%20-%20%20%5Csum_%7Bi=1%7D%5EN%20%5Clog%20p(x_i%20%7C%20s_i)%20+%20R(%5Csss)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannot[ESslashwidehatESleftbrackESslashxESrightbrack, ESslashwidehatESleftbrackESslashsESrightbrack] &= ESslasharg ESslashmin_ESleftbrackESslashx,ESslashsESrightbrack ESslashLy(ESslashx) - ESslashsum_ESleftbracki=1ESrightbrack^N ESslashlog p(x_i | s_i) + R(ESslashsss) ESannotend) /EscapedSubject (ESannoes%5B%5Cwidehat%7B%5Cx%7D,%20%5Cwidehat%7B%5Cs%7D%5D%20%0A&=%20%5Carg%20%5Cmin_%7B%5Cx,%5Cs%7D%20%5CLy(%5Cx)%20-%20%20%5Csum_%7Bi=1%7D%5EN%20%5Clog%20p(x_i%20%7C%20s_i)%20+%20R(%5Csss)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj startxref 69365 %%EOFPK&8E]@originalBimodal.pdf%PDF-1.4 %쏢 5 0 obj <> stream x͜ˎnq=L2 xlK@@: &WZ{wې4Aß,bd3\3}ڮg ^c| 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannot& ESslashmin_ESleftbrackESslashx,ESslashsESrightbrack Q(ESslashx,ESslashx^j) + G(ESslashx|ESslashs)+ R(ESslashsss)= ESslashESslash &ESslashmin_ESleftbrackESslashsESrightbrack ESslashmin_ESleftbrackESslashxESrightbrack ESslashfracESleftbrackLESrightbrackESleftbrack2ESrightbrack ESslashlVert ESslashx - (ESslashx^j - ESslashfracESleftbrack1ESrightbrackESleftbrackLESrightbrack ESslashnabla ESslashLy(ESslashx^j) ) ESslashrVert_2^2 + G(ESslashx|ESslashs)+ R(ESslashsss) ESslashnonumber ESannotend) /EscapedSubject (ESannoes&%20%5Cmin_%7B%5Cx,%5Cs%7D%20Q(%5Cx,%5Cx%5Ej)%20+%20G(%5Cx%7C%5Cs)+%20R(%5Csss)=%20%5C%5C%0A&%5Cmin_%7B%5Cs%7D%20%20%5Cmin_%7B%5Cx%7D%20%20%5Cfrac%7BL%7D%7B2%7D%20%20%5ClVert%20%5Cx%20-%20(%5Cx%5Ej%20-%20%5Cfrac%7B1%7D%7BL%7D%20%5Cnabla%20%5CLy(%5Cx%5Ej)%20%20)%20%5CrVert_2%5E2%20+%20G(%5Cx%7C%5Cs)+%20R(%5Csss)%20%20%5Cnonumber%0A%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(x_i | s_i = 1) = ESslashmathcalESleftbrackQESrightbrack_1ESannotend) /EscapedSubject (ESannoesp(x_i%20%7C%20s_i%20=%201)%20=%20%5Cmathcal%7BQ%7D_1ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack 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ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashsigma_1^2 = 10^2 ESannotend) /EscapedSubject (ESannoes%5Csigma_1%5E2%20=%2010%5E2%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashtextcolorESleftbrackredESrightbrackESleftbrack ESslashLy(ESslashx')+ ESslashlangle ESslashnabla ESslashLy(ESslashx') , ESslashx - ESslashx' ESslashrangle ESrightbrack ESannotend) /EscapedSubject (ESannoes%5Ctextcolor%7Bred%7D%7B%20%5CLy(%5Cx')+%20%5Clangle%20%5Cnabla%20%5CLy(%5Cx')%20,%20%5Cx%20-%20%5Cx'%20%5Crangle%20%7D%0A%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack 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ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannot-ESslashlog p(ESleftbrackESslashbf sESrightbrack) = ESslashsum_ESleftbracki=1ESrightbrack^N ESslashleft (s_i ESslashlog ESslashleft (ESslashfracESleftbrackN-KESrightbrackESleftbrackKESrightbrack ESslashright ) - ESslashlog ESslashleft(1-ESslashfracESleftbrackKESrightbrackESleftbrackNESrightbrack ESslashright) ESslashright ) ESannotend) /EscapedSubject (ESannoes-%5Clog%20p(%7B%5Cbf%20s%7D)%20=%20%5Csum_%7Bi=1%7D%5EN%20%5Cleft%20(s_i%20%5Clog%20%5Cleft%20(%5Cfrac%7BN-K%7D%7BK%7D%20%5Cright%20)%20-%20%5Clog%20%5Cleft(1-%5Cfrac%7BK%7D%7BN%7D%20%5Cright)%20%5Cright%20)%20%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmathcalESleftbrackBESrightbrack(1, ESslashfracESleftbrackKESrightbrackESleftbrackNESrightbrack) ESannotend) /EscapedSubject (ESannoes%5Cmathcal%7BB%7D(1,%20%5Cfrac%7BK%7D%7BN%7D)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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Ნ;UAt!_7wS-Շ=0bIZuy]f݌T_Tz+}?0\B.uP;\l Ϻ!wUh9o24% Nj[,ѝ~NP-OU):90O7[l1fq N p^9`%һEU"Зj&VƠQV[S7N G;}[@$DAK|eRv0k{e>o-*y$D.!B+u5J5kǪϻq:pwF1=P..Ob%XE;-z_8}⡙[nM $)XΖYo q _wOx:UGW.E.y-KH0zU*3GLF'Qvq=>uC;mf$%:iblYUۏIjJXڊ|E7N~l|9D ;e~: }D> endobj 6 0 obj << /Type /ObjStm /N 33 /First 249 /Length 1317 /Filter /FlateDecode >> stream xWnF}W[b&"_vhʒ@Aگ]R"Vھ{9{v83Ԍyτ`J)&S^2 ÜE2&0ei023a5vГ23 8c0NS}L)'xz)exfL  JI)=~WO)CP^3<(:bY;9xz^9/Xzt6]E cI ^^&&N}Y3"ا4SdDTbȎ˯Êc/"g_Cﴛp{1Z0j= dX`xR !m?,B Uؿ+\K9zTsrqx/ \ l\N(); $LB0$쏓OB,:l?B 4$HB5JC:- ]'I&(h}6+>̻'߿?ٻxI~W35BdƲ]i89Q*֣P* l/Eywti#E>)Gowb|Q<|VS%y~ۉ K(lYkViEjET:9, 2 ߵC4Ͷ bhj[!ȍҙF5{d1$Q#lUDC22f%JshM %Ӵ֘5Aڈ 48HNdDË@T;2@O@ABf>.I䓲2 LAWEQ6t˫㏗gߞ+TR^V32`ϫ]JYTaWuWIPN.I^usM/sGT-0+͊Mvg:!ITSF78VT,cXQzNad(p!=e) &qe!N `9UI.˛6EW]ou\UF8*W>IvPVT vhhh9{2[&'GOz{uJ>:7%ygIΰYbs>TtC$\/*Y j.-"?X"\b6Z a!X}k9p)] t endstream endobj 46 0 obj << /Type /XRef /Index [0 47] /Size 47 /W [1 2 1] /Root 44 0 R /Info 45 0 R /ID [ ] /Length 124 /Filter /FlateDecode >> stream x͹ ACQ{]n> : $$b*hSrA$Z?Ȓ~AIAB`;ϔdQE u4D mtP`Ƙ`X`鬔wyQ>_ַ\^N endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashx ESslashsim ESslashexp(-ESslashlambda ESslash| ESslashx ESslash|_0)ESannotend) /EscapedSubject (ESannoes%5Cx%20%5Csim%20%5Cexp(-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_0)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack 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%ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashLy(ESslashx) + G(ESslashx|ESslashs) + R(ESslashsss)ESannotend) /EscapedSubject (ESannoes%5CLy(%5Cx)%20+%20G(%5Cx%7C%5Cs)%20+%20R(%5Csss)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag18.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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'|RBC y+]'Ru m,&]Za0o"Dk%i7>wpnq냎(H kv+ƒs~]]5qq=*:^zYSq7Yf`~$:ڋ^N)'vjMMhOWX7>JJWTlҐIe k GPx_5-{G qErʩك!isH b~#H"ԇ4Bw/4qcnWW' pϯC>BbP{ڙoCVA+Wæs] K49GE|:*]X9W;t[4E?aJ!.tI)j%(MwgW/__QPW8~KS[Z}l(CY,cxqpyvICA6 Tɶad0r'Uۖv,px;훓˃Qf g 4gN6趃 V__Y OrY# <31BB603806F6942AB75C30E7E893FE46>] /Length 167 /Filter /FlateDecode >> stream xϷ DQ{}D$dP 1 Pdv6YDE∄6 [Tyރ˅xD!>!IHA2E(A*PԡMhA:ЅaC&0U"[=8݌Ѹco5VXOb: endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotM^j(ESslashsss) = ESslashsum_ESleftbracki =1ESrightbrack^N ESslashleft ( ESslashfracESleftbrackLESrightbrackESleftbrack2ESrightbrack ESslashBig ( ESslashwidehatESleftbrackxESrightbrack^j_ESleftbracki,ESslashsESrightbrack - (x_i^j - ESslashfracESleftbrack1ESrightbrackESleftbrackLESrightbrack ESslashnabla_i ESslashLy(ESslashx^j) ) ESslashBig )^2 - ESslashlog p(ESslashwidehatESleftbrackxESrightbrack^j_ESleftbracki,ESslashsESrightbrack |s_i) ESslashright ) ESannotend) /EscapedSubject (ESannoesM%5Ej(%5Csss)%20=%20%5Csum_%7Bi%20=1%7D%5EN%20%5Cleft%20(%20%5Cfrac%7BL%7D%7B2%7D%20%5CBig%20(%20%5Cwidehat%7Bx%7D%5Ej_%7Bi,%5Cs%7D%20-%20(x_i%5Ej%20-%20%5Cfrac%7B1%7D%7BL%7D%20%5Cnabla_i%20%5CLy(%5Cx%5Ej)%20%20)%20%5CBig%20)%5E2%20-%20%5Clog%20p(%5Cwidehat%7Bx%7D%5Ej_%7Bi,%5Cs%7D%20%7Cs_i)%20%5Cright%20)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotM^j(ESslashsss) = ESslashsum_ESleftbracki =1ESrightbrack^N ESslashleft ( ESslashfracESleftbrackLESrightbrackESleftbrack2ESrightbrack ESslashBig ( ESslashwidehatESleftbrackxESrightbrack^j_ESleftbracki,ESslashsESrightbrack - (x_i^j - ESslashfracESleftbrack1ESrightbrackESleftbrackLESrightbrack ESslashnabla_i ESslashLy(ESslashx^j) ) ESslashBig )^2 - ESslashlog p(ESslashwidehatESleftbrackxESrightbrack^j_ESleftbracki,ESslashsESrightbrack |s_i) ESslashright ) ESannotend) /EscapedSubject (ESannoesM%5Ej(%5Csss)%20=%20%5Csum_%7Bi%20=1%7D%5EN%20%5Cleft%20(%20%5Cfrac%7BL%7D%7B2%7D%20%5CBig%20(%20%5Cwidehat%7Bx%7D%5Ej_%7Bi,%5Cs%7D%20-%20(x_i%5Ej%20-%20%5Cfrac%7B1%7D%7BL%7D%20%5Cnabla_i%20%5CLy(%5Cx%5Ej)%20%20)%20%5CBig%20)%5E2%20-%20%5Clog%20p(%5Cwidehat%7Bx%7D%5Ej_%7Bi,%5Cs%7D%20%7Cs_i)%20%5Cright%20)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title 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ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is 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ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack 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ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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/Subject (ESannotE = ESslash| ESslashx - ESslashhatESleftbrackESslashxESrightbrackESslash|_2/ESslash| ESslashx ESslash|_2 ESannotend) /EscapedSubject (ESannoesE%20=%20%5C%7C%20%5Cx%20-%20%5Chat%7B%5Cx%7D%5C%7C_2/%5C%7C%20%5Cx%20%5C%7C_2%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) 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(ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack 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%ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotf(ESslashsss) ESannotend) /EscapedSubject (ESannoesf(%5Csss)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack 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0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannots_iESannotend) /EscapedSubject (ESannoess_iESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack 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ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack 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ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmathcalESleftbrackTESrightbrackESannotend) /EscapedSubject (ESannoes%5Cmathcal%7BT%7DESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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/Subject (ESannotp(ESslashs)= ESslashexp(-R(ESslashsss)) ESannotend) /EscapedSubject (ESannoesp(%5Cs)=%20%5Cexp(-R(%5Csss))%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack 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ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashboldsymbolESleftbrackESslashvarepsilonESrightbrack ESslashin ESslashR^MESannotend) /EscapedSubject (ESannoes%5Cboldsymbol%7B%5Cvarepsilon%7D%20%5Cin%20%5CR%5EMESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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Ნ;UAt!_7wS-Շ=0bIZuy]f݌T_Tz+}?0\B.uP;\l Ϻ!wUh9o24% Nj[,ѝ~NP-OU):90O7[l1fq N p^9`%һEU"Зj&VƠQV[S7N G;}[@$DAK|eRv0k{e>o-*y$D.!B+u5J5kǪϻq:pwF1=P..Ob%XE;-z_8}⡙[nM $)XΖYo q _wOx:UGW.E.y-KH0zU*3GLF'Qvq=>uC;mf$%:iblYUۏIjJXڊ|E7N~l|9D ;e~: }D> endobj 6 0 obj << /Type /ObjStm /N 33 /First 249 /Length 1232 /Filter /FlateDecode >> stream xWnF}W[b"~)l!iرTZZleI% wKJ*2;;3{W .A:\kp0p#C2 Mk9!Z]a 4(^{ `ЄF&'|GI)|OrbOzz]i%CqLA)6u'k =B'ͦ0]Nzٲd4 8ώfኡk;]`f'7ogl^Ƿab 8M Gt8[xy`\1{ ^`6~0MJ\0/'l:%,y 1CQ tE‡[M1e[hfӛhh(!>FaWqRa P !wP 6|3awp ɞs݆A;aHCm0kNBaAmqֳ\oYg?f8Uo0xq= ]d+4IXͰRRje!4QVE6GӻI.8\:U駃8a?G/Wfz9͢q6`Ѧ~J?]X8PV$n!e.x0 y8OpМ-:Orrߟals F"q~HEDҚM>_؞ Ҳ@(8?9m`7JߡmP W=MrU (/l\n3ɈK1><{2Đ؅ endstream endobj 46 0 obj << /Type /XRef /Index [0 47] /Size 47 /W [1 2 1] /Root 44 0 R /Info 45 0 R /ID [<511A720E4F7FD87E7332D9A99D520FDC> <511A720E4F7FD87E7332D9A99D520FDC>] /Length 120 /Filter /FlateDecode >> stream x͹ ACQ{Yn> *hM!<َ,ge]A/+vpNUPGMF衏a a;(^吾=WXx endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotQ = ESslash| ESslashhatESleftbrackESslashsESrightbrack - ESslashsESslash|_0 ESannotend) /EscapedSubject (ESannoesQ%20=%20%5C%7C%20%5Chat%7B%5Cs%7D%20-%20%5Cs%5C%7C_0%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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VnQ$ e3w_;M̀ rT*{k搗J$X]߾ oC1һN}1qgj5R MV~*PD-{ImxzOin臇eU]Xӳ  K0siflxFs;]ɨz Y>Kj Lqa_ G4'*t k; x$MZXdjb-RJ#^Hx12;l\-{ְ/=)M@-Dm뤊X m5;EK#Zu" f?ϰ=ƋT%(jq7 zʚTÙfQ@VֺB|7K!GS:/@Li0:1w4cP7~ԃ+?Y.H =3'wfíX_oXCӓ6_#Z##{׭COת֨: Y vү cENDdHaYF5YI,/*ȭBb f3x0dCT&a* u*юWօ5bͭjCvVg;s'BhZ3hʭnwMXD!CY'SG"c{p{I•4I-]NixaB{̝<6cﯪ(XüS1.0Zbum;D;%}[|ol_?P>/"|s=mS,Whkpvڜϕ #gad>0^RsF{,ƖOG,w~iX>\L ތ3UsBl$7_"'wnٷ_*oQD%Gv\\T F@9#>r1Ƶ8;jIOq[pEU4 !NAUP:MS:VqPpMGzS ,F̴4}QGg,"[c0 >#ZT>po0r*?]vvE~ikO'U?vShf拘.b| H*u,Bq}Mt-O$L 2. k3tnv֑pE ~q!=e޹ $/ V!k1<1#J!O <'6Y,w|:?Kk'I>NsDq1yZ&goo/)vVl_YJm+l^꿆0m9HdAE$a0̌ȨYPZU51&O]qG} S]6YWftEHu?c,3F(W~^]U~%Swa&[L:(]KL[álI-FG>TvT)vhߚg,˹R8Arh4쯇W']t1Mԑ-U<Մ6Kq}AHgI3twQx}|U#TFA;0BƄj{s·,K;O]/qhjIJ>뭗VšɔS)GJ-ѮeRm;!s]Y]]YRb[ ג+bfȝ;~_9R^=DNwdH ݧ\=9ap%]vC]nIËLaON: M )Iw 1~q(͝ gͽl"99%Mw" =ٛkG.WuК, >Fz^)TɺK'<|y5Oի|3ciל1 VqB鳵\q$z8㡶/JQnUl||\ִޮ61d`m>MKċU]҉ V 6ޡAhBUE=`FLPEc#*Gm<ٌawl~7kYm qއ<*%]ru,;{i 煐5xuh.jϮ.$%ThX]IJQzK~Le.p//̌<_oM]J٨85 ƹˈTiwoREązj#Fqo+bt[v"= ͗קw], 3hGꥱ!^RNޟN'0Ibm0=׻sbplFD@ 1ml3,#U2a_+p J1kQݏAOC!"E& <դ%k\35#³_W mo+\z:c&E?gKj NZ(l[{kӅY :Yzc5a5>fJXw1.ƍ?Eb1eT&\GgܟE7RA?sJ.eGm[d=`ҌqƬ)&&5~ ;ga TvK)*7-(=_\f}۞[#K0,R83k{8s:SWٖE>]L64֝ )Y|{jݵ}Ԭ#!c)U.1eq٫eq-2bH$!A}zHtJ ecMl,xxBu2>ViYb' fTN{8,eI<@Z#V&Y{!@]&kbyajuZf;߷ZV .}s2o2u ^vy` Ŗs\@ ͇Sj>ܶ}JOMjq,`/VefMj;ƒ~wP鵹w3KfH %e }/^> xZ;]Ġ>3ÛJhTk}#ѽXwrWmrr`7υ3}#HËla\;C+Ko*Yh3"9" |說u~y?3\aZ\.T-Tlɹ$Z5L6c*]Π;ne'6Xf3K e"LSqrkLO6Y1ie11ڒq1CArze{*.16) 8j>}]rbgBA[r5ֹ|@"!nU\fP <S+jw,Ҕx$t =Y8mu{OvҟpŚÜ1exҀxry-6,h<", 9̡6$n-D/AB] xs7XǾgD1l5 Ԃ?Ҋ])PZ]wz.nACQc,^uun]NCw4, p8xwiI0G42EH3w endstream endobj 43 0 obj << /Length1 1380 /Length2 5954 /Length3 0 /Length 6895 /Filter /FlateDecode >> stream xڍVTS]I (]"E:I i!B(M U"A4)RKQHE[wsf9]fb.vkQXQXnh&%`q "(cP,ަ ("HCd`8,##Ԁ#\b@=4 SG{anX|@ "''#;c0( hźÑ0 CAPpbA1(W qS @3/w 4"῅?hWl ^C1@|nWA3; PTtExZb@rEPP):_ExREdM:_@`0~@X"P.$y,Q?Ɵx 2r` ~[y_fCqho+^< ǿ8_?Cq]0,@aǛ ƃWrAq]\oCjj@ NT\(*.B @",Ko 0\@?$/"w#4@, /ωv}^W:í~EӘjtrb6nɺjM'JșOOׄz|>~4&ok*?D/<֧<(h`Q:Wx+I.İH;07x" H(΋? yj!jIa:Nm-Cxhί$Nsv"B"Er }(ت'+g"VVOy=p u/[OȨzxyQ:.\>ޔ?kFյh5)᫰.W*P@.5Dy{@?\Q%UDϡ.;+Π}C! x;np{qZg ""+?®rJƬbSsS3q1 pP$zvfħ;m&>O%3PSf+גކ}I[rClz܎uT"6J j17?xnBuskDr73c{lS[vmf |[鶆(^ ƒPGtϨ{4\'$dښIRv̗4. )%@&?.X/x0fVc?]  ^xx!XvWog.O!C@ 79szU>ӵ$g3=ڂMŽ.Npjڥ}s`\[+a $ی#~\'SˀFX5%]6LWl0zCY3Hf[`C ol"Y]Y.1p8[ @_L RW̏;QFEVGB(pNJSUDFÄ" 5j+<4һӌ&S^jQGE|2*j vj;ׯ~z4y;DiRTBܘÌքvR~a[˷乄^j sFnf!Y3U*!yiaꊛouHM%vCNM-.2uZX{nݴN+6?z zm_uyhrOa0]myӰQ&}^_ -my-P:fMox~]*;TѭIY[=̲x췉|jsmRFFՉmqƑ^{8KOZn?Lk|X6 M~Iyȩpm-eJU BZvS^-WKdiU9rZfhܓR r MR6NY!)-SUΝ { [B6_Dz \mRvS&u\FWm?A!jK(.Uq6#孢7fՀv7^>b!@` uɨPOxn91 E $lzmxՏlzuQb[M|BC5W4&Cmk6ܹo!|F%D`? ka= V_I?Kc.&ɜT]qOH}oA ;jUTuCw7b]7g$g2Yp6K~~H@l˅Dk=# X ~o?}}q$b %YǞRuqe.sPZ=!8s#*a6Zn ၥhtDVW ׽W4N~,?ߟnCkG娊4"NVOpdwl2Ie6zV{%w|3ŊkolokW2!Q7)viu@,m]8i$T[RCNB8"~̒l<`cnvgu-1%_EHo.<)g TyRc>3NFg.-Yw]ohjqp51J>ϛhiZ^Jqg*I^xɭ}QƬEdgl'=^SeWy+17eE>Uyi/Ĉ;#w>7\Gk}稆3 s!ҾWGvX6z?"*ȑ;ۂJPy=*!É W齙c6۸[_gcg.)(l [H_:,CWD$pw.2Ć|:D38c mCճ3 }.e\ɳZ ;6wZ&cm 7p,a?'a,:n YߧA} hgu_!Gk5{v3Wymݍ"nRՖVS1\1]/!>s,̤Tm&l)i߲M {U٘^ f(WæK&d=-X^?Z](DOmxYL񝢠>5)_J ڰQvN3)a`jeTHse@k~bqj%/}Xˋ?S^ >M,C_HJB^!ǘG~[Yeu5~A )PA1 /q.N$S׶_K?E({ʼ{du15flS'Y-l8C d%#= v 4L!" Ნ;UAt!_7wS-Շ=0bIZuy]f݌T_Tz+}?0\B.uP;\l Ϻ!wUh9o24% Nj[,ѝ~NP-OU):90O7[l1fq N p^9`%һEU"Зj&VƠQV[S7N G;}[@$DAK|eRv0k{e>o-*y$D.!B+u5J5kǪϻq:pwF1=P..Ob%XE;-z_8}⡙[nM $)XΖYo q _wOx:UGW.E.y-KH0zU*3GLF'Qvq=>uC;mf$%:iblYUۏIjJXڊ|E7N~l|9D ;e~: }D> endobj 6 0 obj << /Type /ObjStm /N 33 /First 249 /Length 1125 /Filter /FlateDecode >> stream xW[SF~ׯ8oI-}N&3`` a/F]I~+Y[g잳~+Ec✤%XWd% ʼn&mHl ZPЌ4$`@(0JA&ӏ[LA/SK T 7-IL >jA*`ܛi1 {^e6F1ά%'9Owω< jGi0rDO+HւS32]j'_=i3?\ 2K1F nxtv~+p5:y`ڔh\˳'JZ\޸>)"t;-0|B :|;2a05};0soD! )Qx 4 ff]7 n rH*RC__9ڊPcMr&P߭_^>a䗋nm#:tPgWEZ`%p|pbjO.<κ~?UPptFn鞮"1r_-V6-bԽ>;ľ^ jVbSp{V/9)Py)8\8ct+Y\xVGMMmq8<}8)gkp_3m,;%wJoVQ #_Slw:fS279b- endstream endobj 46 0 obj << /Type /XRef /Index [0 47] /Size 47 /W [1 2 1] /Root 44 0 R /Info 45 0 R /ID [<62E8268D5EC801BB1E479CA2213B97C9> <62E8268D5EC801BB1E479CA2213B97C9>] /Length 125 /Filter /FlateDecode >> stream x;aoz[7+qאp gqQ/3SuVP=o߸ ,`w+fRd(*j&ZF]C0S0w[yGt+_??{ endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(x_i | s_i = 0) = ESslashmathcalESleftbrackQESrightbrack_0 ESannotend) /EscapedSubject (ESannoesp(x_i%20%7C%20s_i%20=%200)%20=%20%5Cmathcal%7BQ%7D_0%20ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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U̦[q$Xb_wo6|^v ?m=&Mf`#JW] 4(Ny\6'_/`?Lǥ:; Z'$V˛؛dHwr%5BF?Vb\'nMA=ۛ;'_fՎi-q.(̪@`;~"& f̧nޅ97 i;f~dZYVQ?!cXO)a]vyuZp0ϲzaiι~CQ\Sa[!=Q+u-GLc\X9i& aɴ8|P5:&X[f%ڨ ЋLh+Z6y$DȸdΞ,X;!K. s|ҰR%$ޕ>*]1L!tUTPj`j5]^5dw~PԼ24_ ؗըԎ_ͶN9~Jޤ`+ncRg$I sTT#&"{{#_,5[HTuezOYl{^)=y,'9é+l f)a%#xj';S Tu)׶wz6<=ȽL@ 9최qcţ!nmv1OK}A46K#ˁ'dX6SyGAwN( * ƇAjɄ}@ksL)^Qs:sD,Y[yx OM \dl{ag6N FpTkԨ׸iDZp lRm.Uǥ'k?CV}*`PmwP{6i5D,`9d(cV7Y|S ͠{M#S<9uš3՟IU;QNӅOu!L\lF2r噰^0]Rǖ|դ1Es7$JAMH&7R2B6"MlLw@c VSe^rJ:nVmToj6}eݾMy<6dR4t;`ƾ=OD]3* endstream endobj 31 0 obj << /Length1 1396 /Length2 6062 /Length3 0 /Length 7008 /Filter /FlateDecode >> stream xڍVT$F0 twKI1`6FPIIiEii) %%TiW{{|L𲩅+F!` %7-@ "(f8N#Q@}?o D U)@I0X/G4F G$h @a8B@!0"//';c0( hy07 CqAJ!(@P$VqW p@s8C$@K 7Ga ~(W8H 3Q8 j  H(*r!@mC \ N Erzcфx? u!8n V3B Da>8 !W%k\5H$~a[ qvC\~Ap@!|zzTtpP Ƀp_ <2&A78cp wÿ%tEp@;';A wC&L=rEq=\oс@P\R @$%rC迳/쿵PğɨrC@` ?E h@ho?7o*?P$;O;~8F {OWkKlwE!Ӫ6C %"ڈ@)@̀_8 a`^O 0&8a]R C9IY 'H2@<߼$Ph!H tCc)-xai+@o_`~ a~3_U0vۯ8ėF3Sf#:Tّs}ԾKZB{ӼWw͚CNG× =YWT] 9 "m nFgZ|ХX]>.zbl幬I{%c.qܔ"L;c{LCIЍ"ݼOO-%mvlܤ{LFY?Kf c>9M,R)!P%s%S9dTV x/~%_9W..Ɔ_PRڻ»7ϫ<ͫ˞Vy9DkFMg* GỌ(k0+tP&;Vd'rt,r^#Q6JiwjXJwg}10d~uLpNr=^6AnbδTgu)Li"*5`!S-I[Io+_#y-Uf}Io3MiK/M& =w>;eմkUFSn}ޡT7v>Veg?IX$m+K}3tuv; ?iƅCųx$|T9QeAΓ[ɫ3˖΅. ?47bJZ P}=6l7/lqZCmjnC+"e¿q o?ݥ$']؃>Wws)ԺB~ɼ|hSu jV%jI %~TZOt,mIZճ ׷p<&rݭ}I0ll!rDٌ÷\FE/wN&DOUty66mH~X}B{48Q޺.3B}9 K2!dWѤxa$ѦBJyF}j_o{Z .jR@-njK"R c`iMxPLU=cNû|XTdUd$ /uu2YN"7fgV@| L$ Qxғ-Wydw.gQń~tAqjoeOH< Cv+Z9x[=YOAJ箺fqΉǢE9n):jpQqyԌm:\+N;E#pQ?:>WqNC`aPO t8 m+pٵ!g/LuqpI*f#αY}1zW N >D6Q0<-cg(fL=:'u>S0ޏY0C1vT'*)3mQ /ĕ $k:/Z9UxŚAbYZ1HciF!o)!C9ikʂQ{5pFjўn3z^@/9:?ok/|T!c`m"uHp>ZZ9\1)guonpG]+r&̸5Ř ɴzSJy^5%D+oE<62zM_Т: xAn_aAZ]kuk2~a[7xEu4v_j3G9nߵ@T-B6r3JUV]̺yRCI||ME-7\Vk?޴N(5?~ j +|6]<<.x7 wѼYؘ}m!oϻ>=o|ic`f<,)+rbemn=H̳I}kK3Qy׶rb)_z,?' J]-Oun79( Ђ?:q|r`t5\SnYqP پO-򵔣%+(`)#A.tʵuR_t J'n|{2A$ c&+d%%| 9rw0z}a}yJ+{U2Gek ux$.FDkū&Ԧ~U< 'V{WB+BJPsy'i7vFF26}^g8 [lj7+m#Qb]~VH0w~ͩm;r[Cvшj*.A=c穃[R>}E#W|7ahwVWM7Kkn=$2KI4o .4.OKuԨkQ\;%ƺ&>iH1M,lnC]_|X _ fe>#7~FDIy5M|YXaP4*lKb" x35ܩym.kH9Z&seK$u'NEB}kD>EӦ"2ꧾh5]i}otw|$e}wX?ך|R% .%rS2yʱ.v X ,v$a59z!Pxч:~~wأaw B"nƨ_S2=|IZ 徐mF ԿIv \:8?\ti[y8:Eu-+5YuZ92˕bғ]Xt)fv-[߁ ]#S+}>t< u81hl$baӂiπ% Ԁ ;Br`zûF) )WRsI ЁMq_w.Gʨq=f]j]Moh\ ], 73\Ϳ 5vּCsqgKD߼K~@D >ǘL-8>: >ι{'{}`\+xZ`GPENQ8Cf@ɉk[J O/J~?|2NW66h-5%lS z~ S/9í%]9Z-j[C+vSI"Hao3ɶSmQs",΍ pӳ*=~afN~7r˺ڣnnPM 2نEoĖZmPsג~9IWWTS0.$AŊR[ԘZhD01E& _S я;2{:9;[өn`pk,wh&c_nhF78>Bݶ#@PѢȌ88A1Go9Qյz7 y"QuKde3m>H}v>;,ͧ 3[FuSdL]fn+@0N7͐>,Yo&{}ZT +J/0]@$Ippŭ\3t)s5g&UeR!Jk85֑:lgGs)O.K$*I c> endobj 6 0 obj << /Type /ObjStm /N 23 /First 169 /Length 797 /Filter /FlateDecode >> stream xUO@~_1oJ<^D%QvzHx-HA\ގۏYP ̀`(@\\ZEj>2& Fpz20LxlAhP`܁X?yP>a10JL$gWˉ2v2/Լ\֕ʓEjX%c3BpstT$Z*4zQn`L"72dޑWF('ٔGu3a&TE`ryÓ,f{9k.i~+N_Ze9ۤcn|E4C`Fv׀, tMjУ gO[I[ JUXk5X7aqP;bXn)Q#tTfq*[2Wnrd&Q< zHuZ|c[4M9 nX8k-> wM_oYU ߜt{nkDE0x*4E]CpBa9!4Lx2]:~Ck 9dA3ڷ^U F'&^[l43cAVה2$3$No0M@#@Su دUImk=.=xUqS,W݄{#*BaJD]tȭg}Us]aiۑ1A|x7"iгͰ*gɖg[6ĨBohP< endstream endobj 34 0 obj << /Type /XRef /Index [0 35] /Size 35 /W [1 2 1] /Root 32 0 R /Info 33 0 R /ID [ ] /Length 103 /Filter /FlateDecode >> stream x;PCAE>ڻ@X+aSnZbg4OYZ~q_K7frP`8g4h !O  endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashx^ESleftbrackj+2ESrightbrackESannotend) /EscapedSubject (ESannoes%5Cx%5E%7Bj+2%7DESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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T3"~րi] ~d\Z󫣴M[MѓG`ϟExC#df%9Ka{"}YF,yQ*GE.ԦtG DՌ&B_ag(VB+3*$ (Nff6:tGTn0+ N? w0g/ޜ..z!R EIhܥ*\y\pSٙW1wT8yixmlj1WqMe25I^f5㶷{`5xohk :l-7@qހzo6 #@zfѯ̀X{!!YґέeB endstream endobj 28 0 obj << /Type /XRef /Index [0 29] /Size 29 /W [1 2 1] /Root 26 0 R /Info 27 0 R /ID [<3C5EFE9BE9FB15C71D7BE5A7D8024CB3> <3C5EFE9BE9FB15C71D7BE5A7D8024CB3>] /Length 88 /Filter /FlateDecode >> stream x˻ 0P;!O`zZ:؁qa萘x]G 8$ \۽tB˶?-d$J.RI- 8 endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack 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ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % 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ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashx^jESannotend) /EscapedSubject (ESannoes%5Cx%5EjESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashLy(ESslashx) & ESslashleq ESslashLy(ESslashx') + ESslashlangle ESslashnabla ESslashLy(ESslashx') , ESslashx - ESslashx' ESslashrangle + ESslashfracESleftbrackLESrightbrackESleftbrack2ESrightbrack ESslash|ESslashx - ESslashx'ESslash|_2^2 := ESslashtextcolorESleftbrackredESrightbrackESleftbrackQ(ESslashx,ESslashx')ESrightbrack ESannotend) /EscapedSubject (ESannoes%5CLy(%5Cx)%20&%20%5Cleq%20%5CLy(%5Cx')%20+%20%5Clangle%20%5Cnabla%20%5CLy(%5Cx')%20,%20%5Cx%20-%20%5Cx'%20%5Crangle%20+%20%5Cfrac%7BL%7D%7B2%7D%20%5C%7C%5Cx%20-%20%5Cx'%5C%7C_2%5E2%20%0A:=%20%5Ctextcolor%7Bred%7D%7BQ(%5Cx,%5Cx')%7D%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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T[u v-o !0{BbhC˖Z=SSS?B9IASӴ4rL@&Xbߖ1(.w_&-CcD8dOƷM&d6 jJɳ,ϊIvlͲٴٜIej Z"Ol5˾=Obm6i쨂`אpKZ~޹P4}tuTkWG] -=Q HZnT O΢${c]) AzXiBj pٺXArnq0|1lfHP)/HKbG&h<쮗HFo6 endstream endobj 76 0 obj << /Type /XRef /Index [0 77] /Size 77 /W [1 3 1] /Root 74 0 R /Info 75 0 R /ID [ ] /Length 205 /Filter /FlateDecode >> stream x9NB{d18!3( KĆFX;ƂЙksnwDD*.*#;'/TXWN B ␀S )HCpp9%\AnP2T 5CЂGx6t Ѓ> T?j03ަ󥱞W߾{Q~Zg> endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmathrmESleftbrackproxESrightbrack_ESleftbrackG(ESslashcdot|ESslashs)ESrightbrack(ESslashmathbfESleftbrackuESrightbrack)= ESslasharg ESslashmin_ESleftbrackESslashx ESslashin ESslashreals^NESrightbrack ESslashfracESleftbrack1ESrightbrackESleftbrack2ESrightbrackESslash|ESslashmathbfESleftbrackuESrightbrack - ESslashx ESslash|_2^2 + G(ESslashx|ESslashs) ESannotend) /EscapedSubject (ESannoes%5Cmathrm%7Bprox%7D_%7BG(%5Ccdot%7C%5Cs)%7D(%5Cmathbf%7Bu%7D)=%20%5Carg%20%5Cmin_%7B%5Cx%20%5Cin%20%5Creals%5EN%7D%20%5Cfrac%7B1%7D%7B2%7D%5C%7C%5Cmathbf%7Bu%7D%20-%20%5Cx%20%5C%7C_2%5E2%20+%20%20G(%5Cx%7C%5Cs)%20%0A%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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(ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack 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ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack 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ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack 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%ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashAESslashin ESslashreals^ESleftbrackM ESslashtimes NESrightbrack ESannotend) /EscapedSubject (ESannoes%5CA%5Cin%20%5Creals%5E%7BM%20%5Ctimes%20N%7D%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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F^npl_-Ϳ`hT' _q JQGuϻFɡkrV*[pQυ};`5h2çȟ]k+Ňq!q@Z(6ﳸQ%D`0#o3I> endobj 6 0 obj << /Type /ObjStm /N 38 /First 289 /Length 1272 /Filter /FlateDecode >> stream xW[SF~8oIU'n&(f lJD~GM]xչ|\v {$5iHʑYAҒ BA-1IۖтIA9IU%eb"R$&8P"jϪ¶ؔc0aD+[C840&sa b1 S30}Da -EEP]+<{Rx!dԔI Oڷ̒z%a 6vrʋq^>&cVxL&^ˬQ"yw&H)L-a/Pd К q{cz!} v T.w.mT)y›Oחow%vO@C%S Bl)xFw˞;>d~I?n)mI9TfkFxSYng7fV06S42w7(?9P=G~(Vrp%•2eEFܫ}Q=T8?5fyYG2mF, ,L4F8`5h5_OJB#;!TJ9yƝqWG2y(a>6%ʩGEeCnrJ^.2ìաA J8!{YW>$ݸQ'w.N8W&=! ׈WOA$BX5gl-[#B!b֬٤". |"571: Zohh`,$D6yܹ^8l|cC"F ڮt+PY|K'KƵM _N0yz@K5P/js9_ArTe7lH&fЪTwd endstream endobj 52 0 obj << /Type /XRef /Index [0 53] /Size 53 /W [1 2 1] /Root 50 0 R /Info 51 0 R /ID [ ] /Length 135 /Filter /FlateDecode >> stream xιAa>@@jDB:l%콬"ec {v=d idEyPD eTPEh蠋`Ƙ`|Y~M>K>X˯C7 endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotQ(ESslashx,ESslashx^ESleftbrackj+2ESrightbrack) + G(ESslashx|ESslashs) + R(ESslashsss)ESannotend) /EscapedSubject (ESannoesQ(%5Cx,%5Cx%5E%7Bj+2%7D)%20+%20G(%5Cx%7C%5Cs)%20+%20R(%5Csss)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag18.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack 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ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack 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ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmin_ESleftbrackESslashsss ESslashsubseteq ESslashmathcalESleftbrackNESrightbrackESrightbrack M^j(ESslashsss) + R(ESslashsss)ESannotend) /EscapedSubject (ESannoes%5Cmin_%7B%5Csss%20%5Csubseteq%20%5Cmathcal%7BN%7D%7D%20M%5Ej(%5Csss)%20+%20R(%5Csss)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack 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ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmin_ESleftbrackESslashs ESslashin ESslashESleftbrack 0,1ESslashESrightbrack^NESrightbrack ESslashmin_ESleftbrackESslashx ESslashin ESslashR^NESrightbrack ESslashLy(ESslashx) + G(ESslashx|ESslashs) + R(ESslashsss) ESannotend) /EscapedSubject (ESannoes%5Cmin_%7B%5Cs%20%5Cin%20%5C%7B%200,1%5C%7D%5EN%7D%20%5Cmin_%7B%5Cx%20%5Cin%20%5CR%5EN%7D%20%5CLy(%5Cx)%20+%20G(%5Cx%7C%5Cs)%20+%20R(%5Csss)%0A%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack 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ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotN=1024, ESslashsigma_1=10, ESslashsigma_0=1, p_1=0.9ESannotend) /EscapedSubject (ESannoesN=1024,%20%5Csigma_1=10,%20%5Csigma_0=1,%20p_1=0.9ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag18.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack 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ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotG(ESslashx|ESslashs)= - ESslashsum_ESleftbracki=1ESrightbrack^N ESslashlog p(x_i | s_i) ESannotend) /EscapedSubject (ESannoesG(%5Cx%7C%5Cs)=%20-%20%5Csum_%7Bi=1%7D%5EN%20%5Clog%20p(x_i%20%7C%20s_i)%20%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack 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ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannot-ESslashlog p(ESleftbrackESslashbf sESrightbrack) = ESslashsum_ESleftbracki=1ESrightbrack^N ESslashleft (s_i ESslashlog ESslashleft (ESslashfracESleftbrack1-p_iESrightbrackESleftbrackp_iESrightbrack ESslashright ) - ESslashlog ESslashleft (1-p_i ESslashright ) ESslashright ) ESannotend) /EscapedSubject (ESannoes-%5Clog%20p(%7B%5Cbf%20s%7D)%20=%20%5Csum_%7Bi=1%7D%5EN%20%5Cleft%20(s_i%20%5Clog%20%5Cleft%20(%5Cfrac%7B1-p_i%7D%7Bp_i%7D%20%5Cright%20)%20-%20%5Clog%20%5Cleft%20(1-p_i%20%5Cright%20)%20%5Cright%20)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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/MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack 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ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotx_iESannotend) /EscapedSubject (ESannoesx_iESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashsigma_1^2 = 100^2 ESannotend) /EscapedSubject (ESannoes%5Csigma_1%5E2%20=%20100%5E2%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) 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ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashsssESannotend) /EscapedSubject (ESannoes%5CsssESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotR(ESslashsss)= - ESslashlog p(ESslashs) ESannotend) /EscapedSubject (ESannoesR(%5Csss)=%20-%20%5Clog%20p(%5Cs)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack 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ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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descodscmxlcprt8wtptrXYZ0gXYZDbXYZXrTRClchad|,bTRClgTRCldescGeneric RGB ProfileGeneric RGB Profilemluc skSK(xhrHR(caES$ptBR&ukUA*frFU(Vaeobecn RGB profilGeneri ki RGB profilPerfil RGB genricPerfil RGB Genrico030;L=89 ?@>D09; RGBProfil gnrique RVBu( RGB r_icϏProfilo RGB genericoGenerisk RGB-profil| RGB \ |Obecn RGB profil RGB Allgemeines RGB-Profilltalnos RGB profilfn RGB cϏeNN, RGB 000000Profil RGB generic  RGBPerfil RGB genricoAlgemeen RGB-profielB#D%L RGB 1H'DGenel RGB ProfiliYleinen RGB-profiiliUniwersalny profil RGB1I89 ?@>D8;L RGBEDA *91JA RGB 'D9'EGeneric RGB ProfileGenerel RGB-beskrivelsetextCopyright 2007 Apple Inc., all rights reserved.XYZ RXYZ tM=XYZ Zus4XYZ (6curvsf32 B&l@ExifMM*iC C" }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz w!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ?:V۞lq=*C0 ܃ހWp?y@;yz=~o΀2qښ˻UaPK.HC I'4(Q ot'֤TB G9Hpè6!j?zHU-vYuuWڜ˷;Jt=FpH=4R{U]l 7a<v3[A_F Ӛ㊅Wv{wsEX dc#4ЀleCb!<}hNeۏs3Bqn@{G#Ҥw!(Ƿ5Upn{ 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C}:k|-wݍp]o@D'COiTܛжz~q$Ot_ͪGP|wE"\{&ِy7˃_ gL[-NSasO3>h8Q|̹_XӚZ*O2CՆLCTжkeU<рGqG #L/imֻtcsoJsKݏΠJ*/J&Q5aڄsm@-ǝ.ݸK@pcMؙwfܴ6Tt&+ *>kJ_^eWr'_OҘ4(6 eG.lV'qwr[czij?&;'V[zT5իָ=һyM5l-H9?344ς?3lV@^ožYXQdJDِpIM阘/>`lg)t#Ij&W/2e2߶jQp6)bswFbM/ꙙ?t n-=mV==>( k,4xtrU,O}_UD0N?PL34';-Sif+Brܼ "Ӆq. zD𙟳//. i/s|\ق?T޿vq.FIdpq/SՒ:LۛKЭFI,]!ë6mT*vN\*H#Ŵ[~h|e㑰؊]C+|~=QYe?2rT9>:lbǙs5Zޠ4`mrbޟvcm Ԝ%e._-5ll/CoHf4qm7RQ?Bvܣ3y錎(V!-xfHmiUL+a")\^Y_k@|Ft`NYaCyto+8s|Z'_MBe] yds]$zf_wc N;M1 9䨏ӊ)˪T$e.>x]%e˽Q--?>pVhm-B&* OX YWy"%%4͑M&L&G G0;\fmryP)@ 8BET8Yn,BTG=u?\UTA;Pj~2k+L!G]NޞIhmXWG%[ Λ =Zjr%EV VIBߥ֯ZA'_2g+o]* 'M?赠NL&عz:/H#Y^vݹ=g˳phحaGK^3|=Z<1'!URH^z|pK $J+!)ߩǦm_fJgs]Ng\AhU%3'c)Li65 xq[0YM@"TLJ^d=V\b41A$D {b ywٚ;3[ecrj9jvԉrZ{%6^&5ښKX#v,E٪ɼ<@lQRXV`)c߀l7?}˚&욀0U[J=m'.\.]p)QeڭFȱηfsw}yw5-IY-ևLA@KɅΓb;͇50gsĊјlE/j],|ۥOVEw5)HY2w~r 3"䆨Y}oy<_bʭ[pW0h8JNZEU&b %=;neWGۉ7i!MNb҃iv!&t_ttY,$&*4u *Gun5SqHI?6Bj#{mo{IS<|s>F9_\ X"ҴU] +kITK:xH9x;UCk4e1NJɅcY뜷(9TKBx)LKz^7uSX_ç[V^<[Eԕ%XH=rC{t [|:_7)ÅQO0W2Ӑ< pg6эJQ-Jv<<[G\Ĭm W*ZrrwNp> endobj 6 0 obj << /Type /ObjStm /N 53 /First 409 /Length 2116 /Filter /FlateDecode >> stream xY[SF~ط&bd&@$Ȥ<Svm&ٕKмuVw.+kY`3!R ŔLhfc0g9` {Z &0/ocLj& \aj$)8<t Q\EA'TP0O<(놁n9@\y\!*td[ X_+Lk\agǃ ` De,F>#pu4LbV`Pj +g g VhYY %?> o^v0j I/;)ud*M 8ߟ}b8&4.{^ ҉ɓ(b6.|}B=EfXq<櫂=:<w.;οp6lňe|Ć^=&|:%[nKdMXB5S}ld_MexW0 c<_.U2ھM_P\}T%bek/% ^vzw4)z`XnCgy|"Z<+|2~:U ^8ذu0|xTv5Dh:&Pg|R3$ꖱԺ|9_ ]`P+ﺅoË'`?θpMoiw <011958C79E5E4F010864FFAE7E57701F>] /Length 189 /Filter /FlateDecode >> stream xйmF况 f/)] wЅWa r.miME7";qD%z]m P7WxE 1p! HB Ґ,<%(CP: h3 6t =01L` / 3Kx70ɛjr4U endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashwidehatESleftbrackESslashxESrightbrack^j_ESleftbrackESslashsESrightbrack= ESslashmathrmESleftbrackproxESrightbrack_ESleftbrackG(ESslashcdot|ESslashs)/LESrightbrack(ESslashx^j - ESslashfracESleftbrack1ESrightbrackESleftbrackLESrightbrack ESslashnabla ESslashLy(ESslashx^j)) ESannotend) /EscapedSubject (ESannoes%5Cwidehat%7B%5Cx%7D%5Ej_%7B%5Cs%7D=%20%5Cmathrm%7Bprox%7D_%7BG(%5Ccdot%7C%5Cs)/L%7D(%5Cx%5Ej%20-%20%5Cfrac%7B1%7D%7BL%7D%20%5Cnabla%20%5CLy(%5Cx%5Ej))%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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Co"mfwߛ38]pC_%>F>FDLU "%bHlD$F ёL~kY.-RWLtfcSN0eA9QHNwQDpP}lv0KU noBg}Uk8{,!C;dc0Oߺ2)UQX-+|hrvcǫ^ei,Lk5}+]V\]6R <09E5D9775F171677E81761E2E796EA7A>] /Length 88 /Filter /FlateDecode >> stream x@@d׮K nZ`F 4*(7I1zv/K/K!TRK#- }.;]9 endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp_i ESannotend) /EscapedSubject (ESannoesp_i%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashsss = ESslashESleftbracki | s_i = 1ESslashESrightbrack ESannotend) /EscapedSubject (ESannoes%5Csss%20=%20%5C%7Bi%20%7C%20s_i%20=%201%5C%7D%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 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ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmathcalESleftbrackNESrightbrack(0, ESslashsigma_ESleftbracks_iESrightbrack^2), ESslashsigma_1 > ESslashsigma_0 ESannotend) /EscapedSubject (ESannoes%5Cmathcal%7BN%7D(0,%20%5Csigma_%7Bs_i%7D%5E2),%20%5Csigma_1%20%3E%20%5Csigma_0%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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(ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack 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ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotG(ESslashx|ESslashs) = -ESslashlog p(ESslashx | ESslashs) ESannotend) /EscapedSubject (ESannoesG(%5Cx%7C%5Cs)%20=%20-%5Clog%20p(%5Cx%20%7C%20%5Cs)%0A%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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(ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotN=1024, ESslashlambda=5, ESslashrho=1.09, ESslashsigma_1=10, ESslashsigma_0=1ESannotend) /EscapedSubject (ESannoesN=1024,%20%5Clambda=5,%20%5Crho=1.09,%20%5Csigma_1=10,%20%5Csigma_0=1ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag18.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashsigma_0^2 = 1 ESannotend) /EscapedSubject (ESannoes%5Csigma_0%5E2%20=%201%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack 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).[6_i5%T;[لΊ !Zpv!N~2Gj!C|х*xcD,u'TI:JY4x@BaUK<.-%cD\丢?Aqp('Fvf:;u2|[qAΊ\Ƴ?q;H&򄆧 a( "uk[6hK{)!"`":|mN=IwVlɖ&ΟIQ2#WbI A JUE]6㮛\\ɚr%r> endobj 6 0 obj << /Type /ObjStm /N 28 /First 209 /Length 889 /Filter /FlateDecode >> stream xVnF}W[l"B.A 6+u@Kk, $߳ndeXH\O4E!qNR dĉ"Ph"xDV #ܔ$?F$CoHUHxKl`eJi!fG<.\!Iӑ"6d3Hjs tmI%\ YOBFX}nbqT.wfZDl)|LLA7ްۛY:^smQVdyѹ3 k2CT;Pe+ZɧY4aJu;΍]Ja9Mgrn'?O?@QJB!d߄}>;@PAMQ'\lC0$oC-RvI>Zs/݋Oםկ(zx_Y݉Z6Gܦ~2~# ] /Length 110 /Filter /FlateDecode >> stream xͽ@N_Em(Z aG+'II?{VPނ^K'ao2b 9aXa آ5hpp6^nLg?Z  ) endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashLy(ESslashx) ESannotend) /EscapedSubject (ESannoes%5CLy(%5Cx)%0A%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx)= ESslashfracESleftbrackESslashlambdaESrightbrackESleftbrack2ESrightbrack ESslashexp(ESleftbrack-ESslashlambda ESslash| ESslashx ESslash|_1ESrightbrack)ESannotend) /EscapedSubject (ESannoesp(%5Cx)=%20%5Cfrac%7B%5Clambda%7D%7B2%7D%20%5Cexp(%7B-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1%7D)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx)= ESslashfracESleftbrackESslashlambdaESrightbrackESleftbrack2ESrightbrack ESslashexp(ESleftbrack-ESslashlambda ESslash| ESslashx ESslash|_1ESrightbrack)ESannotend) /EscapedSubject (ESannoesp(%5Cx)=%20%5Cfrac%7B%5Clambda%7D%7B2%7D%20%5Cexp(%7B-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1%7D)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx)= ESslashfracESleftbrackESslashlambdaESrightbrackESleftbrack2ESrightbrack ESslashexp(ESleftbrack-ESslashlambda ESslash| ESslashx ESslash|_1ESrightbrack)ESannotend) /EscapedSubject (ESannoesp(%5Cx)=%20%5Cfrac%7B%5Clambda%7D%7B2%7D%20%5Cexp(%7B-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1%7D)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESannotend) /EscapedSubject (ESannoesESannoesend) /Type (EEtype0EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag0.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx)= ESslashfracESleftbrackESslashlambdaESrightbrackESleftbrack2ESrightbrack ESslashexp(ESleftbrack-ESslashlambda ESslash| ESslashx ESslash|_1ESrightbrack)ESannotend) /EscapedSubject (ESannoesp(%5Cx)=%20%5Cfrac%7B%5Clambda%7D%7B2%7D%20%5Cexp(%7B-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1%7D)ESannoesend) /Type (EEtype0EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag0.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx)= ESslashfracESleftbrackESslashlambdaESrightbrackESleftbrack2ESrightbrack ESslashexp(ESleftbrack-ESslashlambda ESslash| ESslashx ESslash|_1ESrightbrack)ESannotend) /EscapedSubject (ESannoesp(%5Cx)=%20%5Cfrac%7B%5Clambda%7D%7B2%7D%20%5Cexp(%7B-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1%7D)ESannoesend) /Type (EEtype0EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag0.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx)= ESslashfracESleftbrackESslashlambdaESrightbrackESleftbrack2ESrightbrack ESslashexp(ESleftbrack-ESslashlambda ESslash| ESslashx ESslash|_1ESrightbrack)ESannotend) /EscapedSubject (ESannoesp(%5Cx)=%20%5Cfrac%7B%5Clambda%7D%7B2%7D%20%5Cexp(%7B-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1%7D)ESannoesend) /Type (EEtype0EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx)= ESslashfracESleftbrackESslashlambdaESrightbrackESleftbrack2ESrightbrack ESslashexp(ESleftbrack-ESslashlambda ESslash| ESslashx ESslash|_1ESrightbrack)ESannotend) /EscapedSubject (ESannoesp(%5Cx)=%20%5Cfrac%7B%5Clambda%7D%7B2%7D%20%5Cexp(%7B-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1%7D)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx)= ESslashfracESleftbrackESslashlambdaESrightbrackESleftbrack2ESrightbrack ESslashexp(ESleftbrack-ESslashlambda ESslash| ESslashx ESslash|_1ESrightbrack)ESannotend) /EscapedSubject (ESannoesp(%5Cx)=%20%5Cfrac%7B%5Clambda%7D%7B2%7D%20%5Cexp(%7B-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1%7D)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx)= ESslashfracESleftbrackESslashlambdaESrightbrackESleftbrack2ESrightbrack ESslashexp(ESleftbrack-ESslashlambda ESslash| ESslashx ESslash|_1ESrightbrack)ESannotend) /EscapedSubject (ESannoesp(%5Cx)=%20%5Cfrac%7B%5Clambda%7D%7B2%7D%20%5Cexp(%7B-%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1%7D)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESslashx^ESleftbrackj+1ESrightbrack,ESslashs^ESleftbrackj+1ESrightbrack | ESslashy) ESslashgeq p(ESslashx^ESleftbrackjESrightbrack,ESslashs^ESleftbrackjESrightbrack | ESslashy) ESannotend) /EscapedSubject (ESannoesp(%5Cx%5E%7Bj+1%7D,%5Cs%5E%7Bj+1%7D%20%7C%20%5Cy)%20%5Cgeq%20p(%5Cx%5E%7Bj%7D,%5Cs%5E%7Bj%7D%20%7C%20%5Cy)%20%20%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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b;b>xs87K;ۂ-q ?7+ 1Iw}jJy,(~QOx]:e!X>=2wn7^v UI:5|zVk>$6ۖ:<˿f?6ˍ[rʆ~rR LQos-~_voDRQfWsFp%w^Wy9HR影{w #WN9HҒw`s"C_#'"όIPV6S#5 :UsL]2z^|xޘ?{Wq^UTTz/Jf F0fyl9ǚ=l0CP R}p(C5-ܺ\?%^ZJLS)0hhE,?O3[,HL) Ǿ}Ecb0T-A7pF ʻ O{26gwU#+ZbʗsWAhfCM)cʊ=Vu.z@a7]5 i{z}HXiVsvQ'E8jU uܩ/N~LR=3"o4x ?r^Q_g8@<+@}\x6#X.,fq?u%6q H] AҢ> endobj 6 0 obj << /Type /ObjStm /N 53 /First 409 /Length 1942 /Filter /FlateDecode >> stream xXnF}W[&~)lNĵiFbdr},% Ӽ̙ gyτ`J)&S^2 ÜLXa.J lOr$8 mL]b0< H@ĭx=řai$~@?tHx~Kl1xDDfg9f@<iٳ%.\*ŷ\~7໘c^fE>f 8M2xN?dr}W J؛|Ybk1A#荪wҐR3(Xd2_6 9ъ=*7&!{O/Ok+Ulta(zX]D_̓0m1X Fpe ;!DIV4E fM('4zNw-7b.%(TW t%wBjخ7awТ!vލMŦbS6M+B؍MŦb|6M aٍMŦb|6n(A%v#ti4[dUbY, x@QW.1T\q^HNdG`>$'Ȅ꿍D'"WVX/8A:hsJVpCC'4tUyhd5 s4J`*u xJp sEr3!\9g"Pњ,≣lzmX5ʢY4_'5VBS5 U(y!'MuȏJZ5igsәWtf k{voQ2fgq(v\}$  룝:.Ggbr~}hXebt~Qe9ʃ+*iy[Edgc>"VbG:-"O4g WbG!&ڀEJh 9j$)P 2(#upW$H|  FgM;Mǩx(*o_a"Zw ] /Length 189 /Filter /FlateDecode >> stream xлQ8EPPv@@3cFv@@`@#}Ѐ|K~Jҿdҷ)5Yn/ f -0e~Ӕ, #1%(C1T 5 BΠ -8pЁ+.7p ` Vl,OmK^_Ku( endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotR_ESleftbrackESslashtextESleftbrackESslashsc IsingESrightbrackESrightbrack(ESslashs) = ESslashsum_ESleftbrack(i,j) ESslashin ESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashiota(s_i ESslashne s_j) ESannotend) /EscapedSubject (ESannoesR_%7B%5Ctext%7B%5Csc%20Ising%7D%7D(%5Cs)%20=%20%5Csum_%7B(i,j)%20%5Cin%20%5Cmathcal%7BE%7D%7D%20%20%5Ciota(s_i%20%5Cne%20s_j)%20%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmathcalESleftbrackBESrightbrack(1, p_i) ESannotend) /EscapedSubject (ESannoes%5Cmathcal%7BB%7D(1,%20p_i)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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(x3FtL1 PhD!TQ6 FaB vmCkJd Rm zFa(\_UraB ' `7Ie _ &X'LRmm%>qsns؟Z7Ǻ͝Q< ʌGNh#j pp{%* 5u x0sEvҚ@}D=G^G\i_`$&4cQC4[{5ǣz1|(wGhE>)GC8|u*/97Yu7;HJºl#ݬmx-hSmLɉsʹl~¢1Z% 7mI[$mOl${( q$*ZCTpq3IOY4h NBNűKC4ZڏAB( 2YhhLG7HGX:0IƆ.M:?\ ^ڐFoZ@?{GrQRk"5|Ei.s:ANEޤ^׫m٭Nrڨ>yQ_>=<' 再-/wRV/]6&U?}ft5_RA(@0|v^Ŧl<ʘqǦܪ׋|m,ە5#RevIy;hLi0X)AY4bʙuLG,7A5`7yCl>+lnX4-ـN4*Kԋ4RwEԉ58'Vz'lD&"zE>6Onl`dq13:QM[X҂"N>}jbcl +iش7@@ sDGȽ#6jB^o-nޚ;i?o endstream endobj 46 0 obj << /Type /XRef /Index [0 47] /Size 47 /W [1 2 1] /Root 44 0 R /Info 45 0 R /ID [ ] /Length 124 /Filter /FlateDecode >> stream x͹ ACQ{YnJ!!EtA@FF/HH4'ۑ%샬%|vx=Rf(!"GTQC 4B6:袇>b1&bKI>FMk_?? endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotp(ESleftbrackESslashsESrightbrack) ESslashpropto ESslashexp(-ESslashlambda R_ESleftbrackESslashtextESleftbrackESslashsc IsingESrightbrackESrightbrack(ESslashsss) - ESslashrho |ESslashsss|) ESannotend) /EscapedSubject (ESannoesp(%7B%5Cs%7D)%20%5Cpropto%20%5Cexp(-%5Clambda%20R_%7B%5Ctext%7B%5Csc%20Ising%7D%7D(%5Csss)%20-%20%5Crho%20%7C%5Csss%7C)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashtextcolorESleftbrackredESrightbrackESleftbrackO(KESslashlog(K/N))ESrightbrackESannotend) /EscapedSubject (ESannoes%5Ctextcolor%7Bred%7D%7BO(K%5Clog(K/N))%7DESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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K[Å[}N"kG:XFbkcx B*v7@@w<2#"az6 ,j>oQaO fE592,T.94*boя5CxL 6Zdo<[NWUKb?8$9+98Q5֓+Pr?dZͧ ~o6cL/,NsXm+?xٴFJ}_[( б#[82%26wٌ-BE m[{[k1nVit^lGgfy F5ņ=%q!#Zzk=bm*v3n@lP VThSnd͗;:%#95«$nw\EF|k?Ib~}5Q ͅm#:/ץwS sH2+VA.GOYiZ]PbO_yis+-穬l:4--\햋r6DxdZoR =ݙ_ m%Fk@ ?@Ӫ&Yb&EV:6VTn9Q<~شa%F͢{4> endobj 6 0 obj << /Type /ObjStm /N 43 /First 329 /Length 1717 /Filter /FlateDecode >> stream xXr6}W-&/LfEi"oDleILү﹠rH \\ԌgB0)/JDŽa &,2.DmAWlKrc\X$?5y0@B}Xh[ HgJ f'),}L)ǠC, - +qD ftVxZ4C1xra| < xf08o~JϿrdwy+&`R2 Ӽ>yT5plg]q (W-cћ7x{crԻ=/Ey"d ϺpMrrWɅA )/8a)q6=~_$~& }Q2*LL%B ݖAe !;(0!?oF3e&"[C_w'?L$$Y>aW^-0Vzx3]]}6fF҃lܵOEle1쎋IVNw2)r8?{|faPoìa0\> k(X+A(D?j 4Ap~,[7Q ͟QPg BMݴMݴM#֘^miiznWbI촵&Ι2ֹO[+D" >1#-ޝC{ kY[[A>D"91 "I!HW[DGƪkj5x-{Zc+Ue99tjR@e~vU;1mv(QP@|9mٍ9woyqw?>4GKw&Ys-l?|KB9ez{7tŤR~.PyZƻ8hq{B XfK{ueY$l&wZaFEf _u’!,> -d.3 lO &~%J*Xbm&˽|Y gӅ,~:?x$X1lb>"pj^W_qqHw7~ɿssH'[kF8n?CvFTI6&kdQ.@%|3U5x%nn=r[ +B0S,xOb"BLg,CI #8jK CֻYuN&f>Qq-{QWQZ,)R-bVӊt>="+-f쵿>v$GL L'3T7w(b.娟}[˼j`}Z0ysh,l x ]aI,-E4J\,00dP( YғmbTXˤhM=F='5sXE* p^_ZQ1 W|QXLRu xrpd+S >Bt~Pɑ4߭'J!d$H5Yv!G5Љ }Rc-URT*aEgq1+T1vpkGKz Ee4.2R:aM}w3N7MN\>8wNZ 'Y 2s'ZƸAA"?I;hC >&l; )l6V'W,|@=`AAFûVd_D endstream endobj 58 0 obj << /Type /XRef /Index [0 59] /Size 59 /W [1 2 1] /Root 56 0 R /Info 57 0 R /ID [ ] /Length 145 /Filter /FlateDecode >> stream x9o5N)=P9k1G O{3Ӎ%dWs_^QXBr>v>tYGE*:h6:袇>b1bL0 s,{k:L5Φ{~;?3? endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslash| ESslashx ESslash|_0 = ESslashsum_ESleftbracki=1ESrightbrack^N ESslashiota(x_i ESslashneq 0) = KESannotend) /EscapedSubject (ESannoes%5C%7C%20%5Cx%20%5C%7C_0%20=%20%5Csum_%7Bi=1%7D%5EN%20%5Ciota(x_i%20%5Cneq%200)%20=%20KESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmin_ESleftbrackESslashx ESslashin ESslashR^NESrightbrack ESslash| ESslashy - ESslashA ESslashx ESslash|_2^2 + ESslashlambda ESslash| ESslashx ESslash|_1ESannotend) /EscapedSubject (ESannoes%5Cmin_%7B%5Cx%20%5Cin%20%5CR%5EN%7D%20%5C%7C%20%5Cy%20-%20%5CA%20%5Cx%20%5C%7C_2%5E2%20+%20%5Clambda%20%5C%7C%20%5Cx%20%5C%7C_1ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashforall ESslashsss ESslashsubseteq ESslashT ESslashsubseteq ESslashN, ESslashforall e ESslashin ESslashN ESslashsetminus ESslashT, R(ESslashsss ESslashcup ESslashESleftbrackeESslashESrightbrack) - R(ESslashsss) = R(ESslashT ESslashcup ESslashESleftbrackeESslashESrightbrack) - R(ESslashT) ESannotend) /EscapedSubject (ESannoes%5Cforall%20%5Csss%20%5Csubseteq%20%5CT%20%5Csubseteq%20%5CN,%20%5Cforall%20e%20%5Cin%20%5CN%20%5Csetminus%20%5CT,%0A%20R(%5Csss%20%5Ccup%20%5C%7Be%5C%7D)%20-%20R(%5Csss)%20=%20R(%5CT%20%5Ccup%20%5C%7Be%5C%7D)%20-%20R(%5CT)%0A%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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F^npl_-Ϳ`hT' _q JQGuϻFɡkrV*[pQυ};`5h2çȟ]k+Ňq!q@Z(6ﳸQ%D`0#o3I> endobj 6 0 obj << /Type /ObjStm /N 33 /First 249 /Length 1160 /Filter /FlateDecode >> stream xWRF}WBm>V-$a 12[sZ=&&*{9s2#CR $5IM:($-y'H: )0 DYڡT$v$$u }g5)VS{R*Vrx:CX@ 'T| U" G  L5yir(CH"~:ŽфqM6!JZrJβ~ v>A羕_؈Xm,3̳Q3t~o?崵҈TiiPx~gHk yF-Uxv^1DҤciMQ@o9#mc7a ^ , ΡJ`02t p07JZ-z~A®FQea8Z+nG68="g 䃋5PJF#k I1H9f;NaZSrKS4{#htL-[iځv@KaGTrm >n{#a9ʟ/#}.g[a[rvtNolc`>G 26C ݣCu;ufd\4΀MpJLr&~d})y3\D6#V 3hF9`5C4_ͣ߁nMB*qJern=yIɵ5yU7qes &/P9k h.ye Dž\$4T%Rj.g.KsZ)QYVe,WgV{m^8 wR#$Q=#]pwʛ;..8COp7NK`wLlwLZрZ1(( ~-:@UK{?DC/‹ endstream endobj 46 0 obj << /Type /XRef /Index [0 47] /Size 47 /W [1 2 1] /Root 44 0 R /Info 45 0 R /ID [<46A4ACEA58C951BD1615A35A331FE642> <46A4ACEA58C951BD1615A35A331FE642>] /Length 126 /Filter /FlateDecode >> stream xͻQᙻvB$9G5At|gf. ʖw;8yLPB eTPE u4D 9蠋`Ƙ`N?,o&_7{^~9 e endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotQ(ESslashx,ESslashx^j) + G(ESslashx|ESslashs) + R(ESslashsss)ESannotend) /EscapedSubject (ESannoesQ(%5Cx,%5Cx%5Ej)%20+%20G(%5Cx%7C%5Cs)%20+%20R(%5Csss)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag18.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannots_i ESslashin ESslashESleftbrack 0,1ESslashESrightbrack ESannotend) /EscapedSubject (ESannoess_i%20%5Cin%20%5C%7B%200,1%5C%7D%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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[4RNfTpm$E>Ӣߥg["s\ ]ᅰd~\ k7lG*Iy/X0T?QՓ;6qbQhKR7Զ]/R;j=MHf|/Np` =%d~FGqS1+W"s&]FD՞-Db71Eȁ91E7En|Y-ZרT^8&F#ѓeei(61N2P,?ca߼Qdm>9cQʜ}'x$⭬HbpӒ|pA"mgFѩz_3-xb`z6XҪn4`&/|NbHV{[jm;b~U5pkvDoUOW*{"llUX^Jvctȱl9٦=i}B^բ;9! mwM)xMxb1E<ϱ4嵝j&>=YwҚ&vgɧajUA\߹SYi^p%NnR0r3$+cO|\&&Xjp(ؖYұ%Bue.{͡ ؁ Wp nw ֔etm՚>;V׎S&Kp|JZ^n_xM A.Y#ʌ`zж wIW:*;6||t=zIM|3]6'#a1o= QUSHjw&?C!>s3ϫ7fAç su17OߍLqYr?ܖPf]ǾTbƗ=//:Hfv'(D ӒuKYضnORnaTpvbܙ+Lӯvz|VEԋ@k *zHRӎIXN> 7wq8ø{ZoQV8V5٠ge[<&cU\ 89vV H eLz1-)8J|ՇӚ6%AQ)3deꟌ },7f`%pu@P!Uyu]<78}x&'cRAb܍gFu.+^O$LlEzw+5\G'jHŀe ]X[q\EzfN1?)B7Û݈,ٌT4-Lur\\ OvIH| 1\a/Qj"P5$65xj{jԊn5`^4Rη2 >x8l.vՈ^"uHx}%KT{03+ K_PPء 0eBEC&؂)cc]|u=YZl qC,OviS4BsF !E_/ݭJizWLWLyGgTeOiCw3=޿$$\t}q)FƝz:Gl&Ep~5G6XZk@ u/=$E;cyfҾ׽:9n NIq^?Aqp('}Eux->\Fi RV .vO"6N}q{BPq:_,`uOqr’tF"CC6UyVʜ]JrOLYR8ݔvĞ u2}$_. 2\~xHbI :/ ):: 3YngɞZ1|=ri_" 1)»".T%{}C-XD`ҷܔ3VƬG=~`t%~b YƐZ2ϲ=/YG`{j=;'|Rksqcݛ( OH[6.jz>A?bέlL;N3nE7 i`eҵR#KWrP26eH7g!#VOߟۖ^kX` sAi'rHqrm 6-ylr˰VV]wPxpC=pFCvzÌŅi2Ũ2`yJk}1cSC gwKAK@ 60P2yڊ%˥[E>G"*^g]% H`3]LVOY%?[> 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UOBpd#]8@tXtE`P|7:~ @(x'4?! ,:YCb1SaN ]$G:!l{zM\aX@BDDDJF y0_'7M<`f<  `^tgp‘WǛa.ᾀ{`@_oxz9گ1D o򩨠|Aq@PTB$%e5B..(@7$> eoۈH8~xF Pw8O^X,Pxq ;[z0g{P<}aΆpo|y 3Da??6 ~a . ?[#P?'*! @hzI Em@XS< ܨ$yz7//?z9yxb")9GPNrw*"_)_\5ȵdO0`u|#{}G-Wy&Nj+/Eץmq=Q#KFe:'Ofm^q8f3r`YHoRoRR?xΦ Z='hTtxg;@e5Ua4 c̷V>)Bep8dukw[b# zbJya5=À>[.vƲEea߼c Oρ! +К;vW_hxخf;5)"KdӢ*0H"4R>7k<[ {7%=*;7D5 ??w|x{;ML!#7']# XP8Nx{^HЏM7y^b['gFhiw=;G2B%fFqUa]*m2ΫEE*vt_PG%>1*>JE8Y MC9p+ SovJ [ M/N|pZj\ʕC *7.5*ky<3#)Yw3jnbl*fh#n4YrΪ)Ƙ}%_>|쓯FXEx,(CuS !|]"-:Rő,vU$~31Sa-⡚;WMY&tqBU@#n "I,3ooZ'\ dՇ"0I g`yHo19CLVHa iڹ|꣱nْ/UU^+v39(Jѓd_~vWP^<̲{ P`BN2&"㩙*ESJMlI`о#Y+ͬVM]ݲ7uVW[N:⮉̲i=)Ie|8bs yv?cu@0Дnf:*Y VI+:t,>aJB1x%eGI|ZE{ E̻YŶ\N|>|v}-V,\x$OcrgjMz^߄G| Q_UiOv]O Qk`s]*ʿo$7[87:CJ t|,jµZn(VsT7!$ U+cΤ]~&fEA.RdLJPpmUYaW[]<ɋ= Z'nǕ#'`b]oawM]z_۔-JJܧDƟ\ytR[Nl^k&P]4G\DEWxYG)ASRΝ=Dȕ|h"Q7fy !I #xB!` Dӌ >=cNj[[= -.k$;*hV(]3WIPaZ1K0e5 g yeLM_CX$5f 6^=޺ơc½iHpjϸS\F"$ U;:dKky{G|B<^:Z#_ nz.1=?y9f]kQIw+k?# XgOc^%sMD};¼M}5F7fiv4v"9~)<_&jy ݴzij 53T_ByY/#i6S^B^ ǻ r!ݕ\ T}l/q=.&j 0Y=)ojj 0;,QIicBh85V%6,YA4A]8WكrM_Rac^/>; L.z2{!VEΣv1酌|FCnsm\tF)}|acUx'.c@LPfTfv]W~]T{nht%!4]i8:3~ Dz(!R3 (RRBƖ.rZ yxf,:ˡxfd6]K'r#Wh8TK,~ִ,&9m8I<DUNw0ěo\y \*XKh`kdٌ9el%icJUL_[ر@|_gZeVH,09I{A} ew"s؃fnq[0vG,k9I4g_P`JBƭ)!ЩW#ZH),`/ R S+^tM ΢Hw`}տQ=- k{.@qݐ_xj2OwV|Ddɝ*~!yI M l`^k߅(5,~rbݾz_]fS>\(L*Yq \b0; :ٯj#ɮv+@󜴻ћ8芾y+aS7@p$[7ߖZiמ~S5]$t:T"xxnd*fi݄M,]e}W>Qm|qzr2|G@tnKA![o.^J[BFc<A޼p x7{8 6A[: CeWCwb;n&Ӎ\?v&" A?Ҍ`k/tAW%${ MoWOt|u\\RԶ%Q@iѝv0TER |yi,e)/jQ¢SG?hY~Ov- ?xfԫ]B_?7rԞ44aZFʼzq$ qUE1jI'o,qJzW\3$cs񾗛+yZB #+b9;B n)nGb.?[SR'o氨p><3ln\M{8q]%eKUtlCir* )ѓ䇆Arc9to%fW_۬\;=J!fFL.FPiG5$yn~'HQF6]6YGIv%a9,h&_W$ |[ϚlCGWWr!rW xh6FZ;CٚMe7 [ +L,E,FH]j!jn BcML.F"}Ncnר~/; "7okdz38;, Om'6 LꌎP4*EnÖ2p"mn[aM- *1:-1 o^ =ق|1ҔN3}J]ib?y9߹_[6Evɳ+m(=cna $$۩|&11$(zjFެR½IE'}r ) ׄ,֑׮_ -II&y+nr}HWb,D곣KZЏ唝T;DŽ6;=:n2N[,*4xۦ^Q4jf(cݰZG4H̹??Q{KMC"nk= 4wBڈXD!lӵD5\.8dYy jҶuEou% 'z mpX˔ia%U}b仕7mCo@] ~/O`G͜Ydչ8Ig>^3|uN뭥>#{VO4U3)$b>RQ 9 V{$wb)wJjދŪ~~ ZtsvH*̚.*wPE 밿 UxDXh,B?ѩmX02 |@K-k^Hf:rl{޵m:3aEgk ӕʣdiC:YIbD]ʡ~\|5K6 endstream endobj 31 0 obj << /Length1 1371 /Length2 6000 /Length3 0 /Length 6930 /Filter /FlateDecode >> stream xڍwuTTm> RA:paA )$F@$%iPoyoZg{׽24Qc,S,  )LQhJ>3Dz}POM$ G"rA{!5]HVFBN@_د~nA/7E @ QH (ߟR{yPsVâeqįXR$A"(Oה1x&4~-)!{y1Q  8_nxYDžP8 w=^|BF |0OX؍pTY}D%/wS/j2{+&j. UHAZş^:Vw$0pG:.}aK_ʎJj6D/'gDޤTZux6ǞJcvi=R„4ֵn:kJOV[Yt |L !ɥ:cǝ~{ci L+l4j)^j5'&lqkj:5{21ږǂzz7_ʇT/ bn5L cP}pNNē|]*cKlH5xU6)%/N2,ː\z٬ hi,+~v|O{*j{JqyA(?SOk=1Mzw&ӡK?o@ nF56 NgaψTWVUwꨧ| f(j-,vP >+2)3K Z8犐q/)8m\oİiu\4) D!v!ngd]v?{eDa垯/)!XM>Q WڽEyU[ٕEtFV? 3_Ew6N0h,ڥ[fE7Y6aTCȔ$MykG5"EtR/`P+*O GLoߓ7f/KqN'ѕ.g'{j%h+VhpO4j`$)ZQy"0QcAn26cĪi_tIWwߐ:^L߅kͺ[v~s){#WlnZ}obW (i^03׿dZ@פ%1&ɑlIWul-={E=l0N"N| 4m5ͮԮ-\]?)ݼ9 RkouoB#I&GRډ/ų9yv}a Ubd0$͙(_'5qYhOzAl&Fȅc.睌n˹uaв:Y$ްs\LJaK'W.džDafq0Ĉɼ xގ27movI+)b{t<-I ^NjG+9.1J<}l̻4;dۆP2+ M46gJUsnh RD+:xLHˤ@vVt4!rd+I!^(9УiF$ulJjϖ7R)P¯u01η=2a4<(3. y 㫀>xhu;lSQoCR n=t]6#C%z7 ,&v$e`\oA]w=zyLMcBk*;|}VFJ4KYp_ цn 48h5lqj;䝽o$BA(LkFZòZ}2o߅ݘ$$'0SL޽X}_[U}ScQx =_'=W Ut[H6I' >z߬SD-cyI| v&FE= Kbލ}+V.( ;fnj!g'yYZ`6BɞVY؊腳,;dYfv%- !IOt㵣嫹g)zI/;o9 et eIPFf?T;=hj,ɼQ 79@͘HO0Ki77ؽK!x7iۿN/Bit3H8oy}*@\+yç3^ݼ m3qXX-*OY1L8SRYђ 2(dW3̖ ̜Z\1P_eLr9971-gL=.}aHr,Equ%(}nL eȫFFu}5 l|6.c΀99'\XPBTi kt۪r9F}@Ψ  yP6 M gٌa0,j+t cS[۞Zb\#*rV熨HNj:fZ"3/}|]ןϵ l3iLO.b~,EjqkCF~} ;RrJ:]"ݬ ?VNf修}( +Hf>ck_d0(4U+-e&h+ oyy׺׻'X2&#r|\تm<^u|1̺N 8 ;%۪Asѡ$=%8F:rFR.}.tfR%/3 &/5[6~JHv7βΜCI0=3K(g]pmL`]^0v*6l<ۺu;\\ӡeͪcBݱK<2Jam'37>"TL;b*Mgv[[%rYGBIOz䴗L,X„{IO %5 .\$qeBm"|}xY&amøV$8C[ui_"r%,yyNDD#$hkb, ODS59^Зk|ƺ6"T_ "4b*ش`*PMXĤqPʁ[L^+R}5q^pjĦ=wĈw=W*8-a;#һJTLm%H GI(PN {-2;j :.OGb7HC;IWED)J~)xy\Us2 1XѨ6'/(i"{40({qU Dx}O*u&iiJU%БjR0lPf6\2f1mqY1)O+W6G,34+QV%6$!QIeCIj/Ӥf6R' V{ :fG3vdFVCL)/!\rb4$Ql.Jƅ+PdžnP-2>Iy\6!k ٭ωV&`[2,?>PEW=BBܷ쟏yvs<5?ҵFyy>[y){tYvxX,DfL -;: OX"zm& N+'&Q BS{;- 3="g %Fz_30?I{+Mp5¥7noU,-& y{t y ~yicB+$1lS6]uLa ~2;Yxc-BS8;J+X%:NS0"\Vck'[^SѣȠ}v>_FҺRwyOdXfJ|Ћv qEIK@Z*ZS:)o2wYsnHRzyx{HWԲmQP$mfr!}ʿNZϪSg{AꙬSp1AXdbL)YRܑ )" 8ߕJ1i2|?ˡO@\$t\>#H.*U[D.ݾzQ-R%ADK@{>'lGG {[%ޜoN2 O1Ѥ5܎iS`e'xk˓הu s gsem'W6 PxJ6HX*Jwn4a}Л ~wpQѾ(mbIulҭ8b+cI zcNߒ${2 {q&hkG$o{'Ge6 RrsSL+#)h-wmvJ|2,Q1%j/8s sNxTN'olf2X%S΋2xmʩ\R{[7.s );pRY/+J&7ɸO>GEJ'G@;NV:輝kjCӱ-Ί1u:lC?H=i1FUJJ#m7?Ea?LJ "iѩ}$ut~Hi>FFRŘ;\# !qfK+5;,! t{Lu#8%%5タ x |k|{Pznq-ٍ( )var0ȯw0r5o&5zu@$^~Yuo_)l6KJ18)#mRI>Q7$Ut7B-i=Fm zK;V8SsbFF#RF_PQmxO@Ԛ˕kmFB>Y?*8aV)wD,}sTţÞkz,!GDWH$9B*P'B]5nV,GUşk2 6a Ehz4zUE>9ݥm eX)^zu1PTC{q]b58?l mJ@CSyj R*v> endobj 6 0 obj << /Type /ObjStm /N 23 /First 169 /Length 961 /Filter /FlateDecode >> stream xV[o6~ׯ8oMY]P떴i6AS?(6hslO$_HmM?~$#ksR$ Ȉp↬Sġ`A oiq#QDD#"F C&P', JT&,N$K FSKWd})9f>JzEUI+ _.ė$řYw\G.:y'v:@E?<+VL`Ζӧ\FEZ)%l^bA7W`=dE2K¯}EI\qXmҗN0rL}mMcvY ։ %Wt5K͑Ep}SàQ2ȋe%Y;|ԇ1$rRnPfpjśd93?C +!@yBsR 8s"=a7 +Ɯd1屣TkȩhS@Sq Gq(bu\!FFĚAVs[u2[ka` !(;U6''B"NO@GI-~9(ܶN=ķ[`֣w49d!9QibDM[g}ϿvRXK1c?ظ}lrf-@!>Ͳu2T&u˴ ?~~go?d& +N8o'*reӼH8LӃ'LF !}4i^(-S]<{Y΢&d$ endstream endobj 34 0 obj << /Type /XRef /Index [0 35] /Size 35 /W [1 2 1] /Root 32 0 R /Info 33 0 R /ID [<5186800DE0C0B562D803AE29207E734C> <5186800DE0C0B562D803AE29207E734C>] /Length 103 /Filter /FlateDecode >> stream xPD(LUV;]K20q_oIۋ+k5>~' RdQ8 j4h9m߸5M;b endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashy = ESslashA ESslashx + ESslashboldsymbolESleftbrackESslashvarepsilonESrightbrackESannotend) /EscapedSubject (ESannoes%5Cy%20=%20%5CA%20%5Cx%20+%20%5Cboldsymbol%7B%5Cvarepsilon%7DESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashnoise ESslashsim ESslashmathcalESleftbrackNESrightbrack(0,ESslashsigma^2 I) ESannotend) /EscapedSubject (ESannoes%5Cnoise%20%5Csim%20%5Cmathcal%7BN%7D(0,%5Csigma%5E2%20I)%0AESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashmathcalESleftbrackNESrightbrack(0,ESslashsigma_ESleftbracki,s_iESrightbrack), ESslashsigma_ESleftbracki,1ESrightbrack > ESslashsigma_ESleftbracki,0ESrightbrackESannotend) /EscapedSubject (ESannoes%5Cmathcal%7BN%7D(0,%5Csigma_%7Bi,s_i%7D),%20%5Csigma_%7Bi,1%7D%20%3E%20%5Csigma_%7Bi,0%7DESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotG(ESslashx|ESslashs)ESannotend) /EscapedSubject (ESannoesG(%5Cx%7C%5Cs)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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F^npl_-Ϳ`hT' _q JQGuϻFɡkrV*[pQυ};`5h2çȟ]k+Ňq!q@Z(6ﳸQ%D`0#o3I> endobj 6 0 obj << /Type /ObjStm /N 38 /First 289 /Length 1260 /Filter /FlateDecode >> stream xWRH}Wܷ ޭTL@B-@]I1?J^{p2y*h[w9:}nC $5IM:($-y'H: aA(:lLW~RrM[#n=݌)/FGx>Mq:Ո]X*Y?+:|4!Gݴgjp ySv!>7a jA!]@!a$5Z`h0P`lC00u'D/nI2Vu"qA¬NSNSN3+rB,+.$KAh<8 r5BS}o.?\~ywrr$^z*ݲW6d[&0z􌝢S t>'SXutNѶ8{<':>?gqgnU9䍙en混+()Hvp]mtWl{6#y gkBMJLr"SCսv\=T<=Kr˴t2aX 3sj 1hy3>[GG/݋Z7!T\+sFy;7nyJWߌ/e5L)8, r%fn4?L|V:⓯+qGZ71:О'+b/%ߠ}ka" RJ)@; #eK.I([\]ܿRETMb,ڼ":l7ͭ˵vw50]ov]|_ZM:Zڔٙm :n! _ yZ_ hFEM(@vw5ɽ ,  endstream endobj 52 0 obj << /Type /XRef /Index [0 53] /Size 53 /W [1 2 1] /Root 50 0 R /Info 51 0 R /ID [<938F0BD34D41AC0D45969FCC5F005F02> <938F0BD34D41AC0D45969FCC5F005F02>] /Length 135 /Filter /FlateDecode >> stream xλ;yNdF j0:P@*QfH! l%콬es ;݁{!2EyPD eTPEh蠋`Ƙ`掾rrpO;:H?X˯; endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotQ(ESslashx,ESslashx^ESleftbrackj+1ESrightbrack) + G(ESslashx|ESslashs) + R(ESslashsss)ESannotend) /EscapedSubject (ESannoesQ(%5Cx,%5Cx%5E%7Bj+1%7D)%20+%20G(%5Cx%7C%5Cs)%20+%20R(%5Csss)ESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag18.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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(ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack 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ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack 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/Subject (ESannotx_iESannotend) /EscapedSubject (ESannoesx_iESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) 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s`OZVON"NxBN9$<]Viv=-iG{v'ϥYX\VSm@ԕv^ehu?flY$2o5ZQk zGWsTd޿S^GY3~ikCnZsWjڜ#U Zq8'㣵51Q2}OաOH'}ҥ Ή5kD̔{?24v2M=zT|˙c րr\Jv`_&*R N;ԄS,J:+Wm_MK4~o+ؕ1݇xԃ@M;:3`BD^eLg%_UTtO^}QH 2JV$$2м3#7|4p| ωxЇ;a&͆)> endobj 6 0 obj << /Type /ObjStm /N 23 /First 169 /Length 800 /Filter /FlateDecode >> stream xUmO0_qFŚ؎ !і[ IMV(I;'}I;o$g=ݙm36>e.P<-|z&񁟌 |s =C&<`6 4D(0܁X!C||\ )Ijg˩2v*.rp׆ud*ZI$[+.8Hak%E28H'Cuפ&'&%SHpWfY(8|ar"M9gOK& 3jV 9?<3| Xs΢Wq 7*15&m2½fEfd b+ /@פ&i}wųVh=*`d;VkͰ4"5 =jG +m4A\2e1j,JPArKJoV*N0D+tOĭ֒v[4ۡ]ipyMyK%H%);l6h|ƝUkQG+V9d/r$ `L[*ΖΣtvM!QxO 0>nQ "tbbb6YsE gh#]S븒{ 8z;84 JM}Jx+`zzZ {^ X~;0s?oN=aD]$VV:_[:ѵNu~ NP { [7#`Fl.\ = [4A^lK&UFXQO% endstream endobj 34 0 obj << /Type /XRef /Index [0 35] /Size 35 /W [1 2 1] /Root 32 0 R /Info 33 0 R /ID [ ] /Length 103 /Filter /FlateDecode >> stream x;PCAE>ڻ{Zfhp%n8X WIDߞd} ,1ȐGpԸAlW~ endstream endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%7Balgorithm%7D%0A%5Cusepackage%7Balgorithmic%7D%0A%5Cusepackage%7Burl%7D%0A%5Cusepackage%7Bamsthm%7D%20%20%25%20defines%20the%20'proof'%20environment%20with%20square%20symbol%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%7Benumerate%7D%0A%0A%5Cusepackage%5Bfont=small,textfont=sl%5D%7Bcaption%7D%0A%5Cusepackage%7Benumitem%7D%0A%0A%5Cnewcommand%7B%5Cnew%7D%7B%5Cnewcommand%7D%0A%5Cnew%7B%5CV%7D%7B%5Cmathcal%7BV%7D%7D%0A%5Cnew%7B%5CE%7D%7B%5Cmathcal%7BE%7D%7D%0A%5Cnew%7B%5Cscal%7D%5B2%5D%7B%5Cleft%5Clangle%7B%231%7D,%7B%232%7D%5Cright%5Crangle%7D%0A%0A%20%0A%5CDeclareCaptionType%7Bcopyrightbox%7D%0A%5Cnewcommand%7B%5Cdenselist%7D%7B%5Citemsep%200pt%5Ctopsep-6pt%5Cpartopsep-6pt%7D%0A%0A%5Cnewcommand%7B%5Cvectornorm%7D%5B1%5D%7B%5C%7C%231%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormbig%7D%5B1%5D%7B%5CBig%5C%7C%231%5CBig%5C%7C%7D%0A%5Cnewcommand%7B%5Cvectornormmed%7D%5B1%5D%7B%5Cbig%5C%7C%231%5Cbig%5C%7C%7D%0A%0A%5Cnewcommand%7B%5Cobservation%7D%7Bw%7D%20%25%20this%20used%20to%20be%20w%20when%20signal%20was%20beta%0A%5Cnewcommand%7B%5Cobs%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Csensing%7D%7B%7B%5Cbf%20%5CPhi%7D%7D%0A%5Cnewcommand%7B%5Crow%7D%7B%7B%5Cboldsymbol%7B%5Cphi%7D%7D%7D%0A%5Cnewcommand%7B%5Csignal%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5CAst%7D%7B%5Cstar%7D%20%25%20use%20either%20%5Cstar%20or%20%5Cast%0A%25%20%5Cnewcommand%7B%5Clp%7D%7B%5Cast%7D%20%25%20for%20%7C%7Cx%7C%7C_%5Clp,%20but%20don't%20use%20p%20since%20that%20is%20dimension%0A%5Cnewcommand%7B%5Clp%7D%7Bq%7D%20%25%0A%5Cnewcommand%7B%5Cbestsignal%7D%7B%5Csignal%5E%5CAst%7D%0A%5Cnewcommand%7B%5Cnoise%7D%7B%5Cvarepsilon%7D%0A%5Cnewcommand%7B%5Cdimension%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cnumsam%7D%7Bm%7D%0A%5Cnewcommand%7B%5Csparsity%7D%7Bs%7D%0A%0A%5Cnewcommand%7B%5Cx%7D%7B%7B%5Cbf%20x%7D%7D%0A%5Cnewcommand%7B%5Cy%7D%7B%7B%5Cbf%20y%7D%7D%0A%5Cnewcommand%7B%5Cs%7D%7B%7B%5Cbf%20s%7D%7D%0A%5Cnewcommand%7B%5Cz%7D%7B%7B%5Cbf%20z%7D%7D%0A%5Cnewcommand%7B%5CY%7D%7B%7B%5Cbf%20Y%7D%7D%0A%0A%5Cnew%7B%5CA%7D%7B%7B%5Cbf%20A%7D%7D%0A%5Cnew%7B%5Csss%7D%7B%5Cmathcal%7BS%7D%7D%0A%5Cnew%7B%5CT%7D%7B%5Cmathcal%7BT%7D%7D%0A%5Cnew%7B%5CN%7D%7B%5Cmathcal%7BN%7D%7D%0A%5Cnew%7B%5CbigO%7D%7B%5Cmathcal%7BO%7D%7D%0A%5Cnew%7B%5CLy%7D%7B%5Cmathcal%7BL%7D_%7By%7D%7D%0A%5Cnewcommand%7B%5Ctr%7D%7B%5Ctext%7Btr%7D%7D%0A%0A%25%5Cnewcommand%7B%5Cconstraint%7D%7B%5Cmathcal%7BC%7D_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstraint%7D%7B%5CSigma_%5Csparsity%7D%0A%5Cnewcommand%7B%5Cconstrainttwo%7D%7B%5Cmathcal%7BC%7D_%7B2%5Csparsity%7D%7D%0A%5Cnewcommand%7B%5Cstructsub%7D%7B%5Cmathcal%7BV%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Cprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%7D%0A%5Cnewcommand%7B%5Capprprojection%7D%7B%5Cmathcal%7BP%7D_%7B%5Cconstraint%7D%5E%7B%5Cepsilon%7D%7D%0A%0A%0A%5Cnewcommand%7B%5CconvexSimple%7D%7B%5Cmathcal%7BK%7D%7D%0A%5Cnewcommand%7B%5Cconvex%7D%7B%5CconvexSimple_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cnonconvex%7D%7B%5Cconstraint%7D%0A%5Cnewcommand%7B%5Cball%7D%7B%5Cmathcal%7BB%7D_%7B%5Clambda%7D%7D%0A%5Cnewcommand%7B%5Cproj%7D%7B%5Cmathcal%7BP%7D%7D%0A%5Cnewcommand%7B%5Csupp%7D%7B%5Cmathrm%7Bsupp%7D%7D%0A%5Cnewcommand%7B%5Cprox%7D%7B%5Cmathrm%20prox%7D%0A%5Cnewcommand%7B%5Csofth%7D%7B%5Cmathrm%20Soft%7D%0A%5Cnewcommand%7B%5Cargminb%7D%7B%5Cmathrm%20argmin%7D%0A%25%20for%20ones,%20we%20can%20also%20use%20this:%20%5Cmathbf%7B1%7D%0A%5Cnewcommand%7B%5Cbbfamily%7D%7B%5Cfontencoding%7BU%7D%5Cfontfamily%7Bbbold%7D%5Cselectfont%7D%0A%5Cnewcommand%7B%5Ctextbb%7D%5B1%5D%7B%7B%5Cbbfamily%231%7D%7D%0A%5CDeclareMathAlphabet%7B%5Cmathbbb%7D%7BU%7D%7Bbbold%7D%7Bm%7D%7Bn%7D%0A%5Cnewcommand%7B%5Cones%7D%7B%5Cmathbbb%201%7D%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%25%20ones%20vector%0A%5Cnewcommand%7B%5Corder%7D%7B%5Cmathcal%7BO%7D%7D%0A%0A%5Cnewcommand%7B%5Cvect%7D%5B2%5D%7B%20%5Cbegin%7Bbmatrix%7D%20%231%20%5C%5C%20%232%20%5Cend%7Bbmatrix%7D%7D%0A%0A%5Cnewcommand%7B%5CBlackBox%7D%7B%5Crule%7B1.5ex%7D%7B1.5ex%7D%7D%20%20%25%20end%20of%20proof,%20or%20use%20amsthm%20environments%0A%25%5Cnewcommand%7B%5Cargmax%7D%7B%5Carg%5Cmax%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmax%7D%7Barg%5C,max%7D%0A%25%5CDeclareMathOperator*%7B%5Cargmin%7D%7Barg%5C,min%7D%0A%25%5CDeclareMathOperator*%7B%5Cargminmax%7D%7Barg%5C,min%5Csetminus%20max%7D%0A%0A%0A%25---%0A%5Cnewcommand%7B%5Cqq%7D%7B%5Cvspace*%7B-3mm%7D%7D%0A%5Cnewcommand%7B%5CR%7D%7B%20%5Cmathbb%7BR%7D%20%7D%20%20%20%20%20%25%7B%7B%5Cbf%20R%7D%7D%0A%5Cnewcommand%7B%5Creals%7D%7B%5CR%7D%0A%5Cnewtheorem%7Blemma%7D%7BLemma%7D%0A%5Cnewtheorem%7Bprop%7D%7BProposition%7D%0A%5Cnewtheorem%7Bdefinition%7D%7BDefinition%7D%0A%5Cnewtheorem%7Bremark%7D%7BRemark%7D%0A%5Cnewtheorem%7Btheorem%7D%7BTheorem%7D%0A%5Cnewtheorem%7Bcorollary%7D%7BCorollary%7D%0A%5Cnewtheorem%7Bconjecture%7D%7BConjecture%7D%0A%5Cnewtheorem%7Bproblem%7D%7BProblem%7D%0A%0A%0AESannoepend) /Subject (ESannotESslashx^ESleftbrackj+1ESrightbrackESannotend) /EscapedSubject (ESannoes%5Cx%5E%7Bj+1%7DESannoesend) /Type (EEtype4EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag28.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackageESleftbrackalgorithmESrightbrack ESslashusepackageESleftbrackalgorithmicESrightbrack ESslashusepackageESleftbrackurlESrightbrack ESslashusepackageESleftbrackamsthmESrightbrack % defines the 'proof' environment with square symbol ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackageESleftbrackenumerateESrightbrack ESslashusepackage[font=small,textfont=sl]ESleftbrackcaptionESrightbrack ESslashusepackageESleftbrackenumitemESrightbrack ESslashnewcommandESleftbrackESslashnewESrightbrackESleftbrackESslashnewcommandESrightbrack ESslashnewESleftbrackESslashVESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrackESrightbrack ESslashnewESleftbrackESslashEESrightbrackESleftbrackESslashmathcalESleftbrackEESrightbrackESrightbrack ESslashnewESleftbrackESslashscalESrightbrack[2]ESleftbrackESslashleftESslashlangleESleftbrack#1ESrightbrack,ESleftbrack#2ESrightbrackESslashrightESslashrangleESrightbrack ESslashDeclareCaptionTypeESleftbrackcopyrightboxESrightbrack ESslashnewcommandESleftbrackESslashdenselistESrightbrackESleftbrackESslashitemsep 0ptESslashtopsep-6ptESslashpartopsep-6ptESrightbrack ESslashnewcommandESleftbrackESslashvectornormESrightbrack[1]ESleftbrackESslash|#1ESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormbigESrightbrack[1]ESleftbrackESslashBigESslash|#1ESslashBigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashvectornormmedESrightbrack[1]ESleftbrackESslashbigESslash|#1ESslashbigESslash|ESrightbrack ESslashnewcommandESleftbrackESslashobservationESrightbrackESleftbrackwESrightbrack % this used to be w when signal was beta ESslashnewcommandESleftbrackESslashobsESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsensingESrightbrackESleftbrackESleftbrackESslashbf ESslashPhiESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrowESrightbrackESleftbrackESleftbrackESslashboldsymbolESleftbrackESslashphiESrightbrackESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsignalESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashAstESrightbrackESleftbrackESslashstarESrightbrack % use either ESslashstar or ESslashast % ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackESslashastESrightbrack % for ||x||_ESslashlp, but don't use p since that is dimension ESslashnewcommandESleftbrackESslashlpESrightbrackESleftbrackqESrightbrack % ESslashnewcommandESleftbrackESslashbestsignalESrightbrackESleftbrackESslashsignal^ESslashAstESrightbrack ESslashnewcommandESleftbrackESslashnoiseESrightbrackESleftbrackESslashvarepsilonESrightbrack ESslashnewcommandESleftbrackESslashdimensionESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashnumsamESrightbrackESleftbrackmESrightbrack ESslashnewcommandESleftbrackESslashsparsityESrightbrackESleftbracksESrightbrack ESslashnewcommandESleftbrackESslashxESrightbrackESleftbrackESleftbrackESslashbf xESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashyESrightbrackESleftbrackESleftbrackESslashbf yESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsESrightbrackESleftbrackESleftbrackESslashbf sESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashzESrightbrackESleftbrackESleftbrackESslashbf zESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashYESrightbrackESleftbrackESleftbrackESslashbf YESrightbrackESrightbrack ESslashnewESleftbrackESslashAESrightbrackESleftbrackESleftbrackESslashbf AESrightbrackESrightbrack ESslashnewESleftbrackESslashsssESrightbrackESleftbrackESslashmathcalESleftbrackSESrightbrackESrightbrack ESslashnewESleftbrackESslashTESrightbrackESleftbrackESslashmathcalESleftbrackTESrightbrackESrightbrack ESslashnewESleftbrackESslashNESrightbrackESleftbrackESslashmathcalESleftbrackNESrightbrackESrightbrack ESslashnewESleftbrackESslashbigOESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewESleftbrackESslashLyESrightbrackESleftbrackESslashmathcalESleftbrackLESrightbrack_ESleftbrackyESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashtrESrightbrackESleftbrackESslashtextESleftbracktrESrightbrackESrightbrack %ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstraintESrightbrackESleftbrackESslashSigma_ESslashsparsityESrightbrack ESslashnewcommandESleftbrackESslashconstrainttwoESrightbrackESleftbrackESslashmathcalESleftbrackCESrightbrack_ESleftbrack2ESslashsparsityESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashstructsubESrightbrackESleftbrackESslashmathcalESleftbrackVESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashapprprojectionESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrack_ESleftbrackESslashconstraintESrightbrack^ESleftbrackESslashepsilonESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexSimpleESrightbrackESleftbrackESslashmathcalESleftbrackKESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashconvexESrightbrackESleftbrackESslashconvexSimple_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashnonconvexESrightbrackESleftbrackESslashconstraintESrightbrack ESslashnewcommandESleftbrackESslashballESrightbrackESleftbrackESslashmathcalESleftbrackBESrightbrack_ESleftbrackESslashlambdaESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashprojESrightbrackESleftbrackESslashmathcalESleftbrackPESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashsuppESrightbrackESleftbrackESslashmathrmESleftbracksuppESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashproxESrightbrackESleftbrackESslashmathrm proxESrightbrack ESslashnewcommandESleftbrackESslashsofthESrightbrackESleftbrackESslashmathrm SoftESrightbrack ESslashnewcommandESleftbrackESslashargminbESrightbrackESleftbrackESslashmathrm argminESrightbrack % for ones, we can also use this: ESslashmathbfESleftbrack1ESrightbrack ESslashnewcommandESleftbrackESslashbbfamilyESrightbrackESleftbrackESslashfontencodingESleftbrackUESrightbrackESslashfontfamilyESleftbrackbboldESrightbrackESslashselectfontESrightbrack ESslashnewcommandESleftbrackESslashtextbbESrightbrack[1]ESleftbrackESleftbrackESslashbbfamily#1ESrightbrackESrightbrack ESslashDeclareMathAlphabetESleftbrackESslashmathbbbESrightbrackESleftbrackUESrightbrackESleftbrackbboldESrightbrackESleftbrackmESrightbrackESleftbracknESrightbrack ESslashnewcommandESleftbrackESslashonesESrightbrackESleftbrackESslashmathbbb 1ESrightbrack % ones vector ESslashnewcommandESleftbrackESslashorderESrightbrackESleftbrackESslashmathcalESleftbrackOESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashvectESrightbrack[2]ESleftbrack ESslashbeginESleftbrackbmatrixESrightbrack #1 ESslashESslash #2 ESslashendESleftbrackbmatrixESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashBlackBoxESrightbrackESleftbrackESslashruleESleftbrack1.5exESrightbrackESleftbrack1.5exESrightbrackESrightbrack % end of proof, or use amsthm environments %ESslashnewcommandESleftbrackESslashargmaxESrightbrackESleftbrackESslashargESslashmaxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargmaxESrightbrackESleftbrackargESslash,maxESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminESrightbrackESleftbrackargESslash,minESrightbrack %ESslashDeclareMathOperator*ESleftbrackESslashargminmaxESrightbrackESleftbrackargESslash,minESslashsetminus maxESrightbrack %--- ESslashnewcommandESleftbrackESslashqqESrightbrackESleftbrackESslashvspace*ESleftbrack-3mmESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashRESrightbrackESleftbrack ESslashmathbbESleftbrackRESrightbrack ESrightbrack %ESleftbrackESleftbrackESslashbf RESrightbrackESrightbrack ESslashnewcommandESleftbrackESslashrealsESrightbrackESleftbrackESslashRESrightbrack ESslashnewtheoremESleftbracklemmaESrightbrackESleftbrackLemmaESrightbrack ESslashnewtheoremESleftbrackpropESrightbrackESleftbrackPropositionESrightbrack ESslashnewtheoremESleftbrackdefinitionESrightbrackESleftbrackDefinitionESrightbrack ESslashnewtheoremESleftbrackremarkESrightbrackESleftbrackRemarkESrightbrack ESslashnewtheoremESleftbracktheoremESrightbrackESleftbrackTheoremESrightbrack ESslashnewtheoremESleftbrackcorollaryESrightbrackESleftbrackCorollaryESrightbrack ESslashnewtheoremESleftbrackconjectureESrightbrackESleftbrackConjectureESrightbrack ESslashnewtheoremESleftbrackproblemESrightbrackESleftbrackProblemESrightbrack ESannopend) /EscapedPreamble 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