EPFL Applied Topology Seminar 2020/21

The seminar will take place online until further notice. If you wish to participate please contact Adélie Garin , Celia Hacker or Kathryn Hess to receive the zoom link.

Program

Date and place Title Speaker
22.09.20,
15:15
A directed persistent homology theory for dissimilarity functions. David Méndez
University of Southampton
29.09.20,
16.15
Gromov-Wasserstein in a Riemannian framework with applications to neuroimaging Samir Chowdhury
Stanford
06.10.20,
15.15
Stochastic dynamics on CW complexes Michael Catanzaro
Iowa State University
03.11.20,
15.15
Persistent Stiefel-Whitney classes Raphaël Tinarrage
Inria
10.11.20,
15.15
Probabilistic Convergence and Stability of Random Mapper Graphs Bei Wang
University of Utah
08.12.20,
15.15
TBA Nicolas Berkouk
EPFL

Abstracts

D. Méndez:  Persistent homology is one of the most successful tools in Topological Data Analysis, having been applied in numerous scientific domains such as medicine, neuroscience, robotics, and many others. However, a fundamental limitation of persistent homology is its inability to incorporate directionality.
In this talk we will introduce a theory of persistent homology for directed simplicial complexes which detects directed cycles in odd dimensions. To do so, we introduce a homology theory with coefficients in semirings for these complexes: by explicitly removing additive inverses, we can detect directed cycles algebraically. We will also exhibit some of the features of this persistent homology theory, including its stability and how the obtained persistent diagrams relate to those obtained from persistent homology with ring coefficients. We will end the talk by highlighting some of the computational challenges towards the effective computation of the directed persistent diagram of a pointcloud.

S. Chowdhury:  Geometric and topological data analysis methods are increasingly being used in human neuroimaging studies to derive insights into neurobiology and behavior. We will begin by describing a pipeline that utilizes the Mapper algorithm to produce network representations of whole-brain activity during ongoing cognition. When applying this pipeline at scale across clinical populations, however, generating consistent insights requires the development of statistical learning techniques such as averaging and PCA across graphs without known node correspondences. We formulate this problem using the Gromov-Wasserstein (GW) distance and present a recently-developed Riemannian framework for GW-based graph averaging, partitioning, and tangent PCA. This framework permits using derived network representations beyond graph geodesic distances or adjacency matrices. In particular, we show that compared to state-of-the-art implementations that use adjacency matrix formulations, a spectral network representation leads to improved accuracy and runtime in graph learning tasks. Additionally, we observe that the spectral approach to GW graph partitioning corresponds to a generalization of Fiedler bipartitioning, thus suggesting new avenues for rigorous analysis of the GW problem.

M. Catanzaro: In this talk, we will explore stochastic motion of subcomplexes inside CW complexes. We refer to this as a Markov CW-chain and it serves as a generalization of a random walks on a graph, and a discretization of stochastic flows on smooth manifolds.
We will define a notion of stochastic current, connect it to classical electric current, and show it satisfies quantization. Along the way, we will define the main combinatorial objects of study, namely spanning trees and spanning co-trees in higher dimensions,and relate these to the dynamics.

R. Tinarrage: Persistent homology can be seen as an answer to the following estimation problem: given a finite sample of a nice subset of the Euclidean space, estimate the homology groups of the nice subset. By nice subset we mean a C^2-submanifold, or more generally a positive-reach subset, and by homology groups we mean singular homology groups over a finite field. However, in algebraic topology, there exists many other topological invariants than homology groups. In this talk, we will deal with Stiefel-Whitney classes. These classes are associated to any topological space endowed with a vector bundle structure, and they carry more topological information than the cohomology groups alone. Our new estimation problem is the following: given a finite sample of a vector bundle, estimate the Stiefel-Whitney classes of the vector bundle. We will adopt a persistent approach.

B. Wang: Persistent homology can be seen as an answer to the following estimation problem: given a finite sample of a nice subset of the Euclidean space, estimate the homology groups of the nice subset. By nice subset we mean a C^2-submanifold, or more generally a positive-reach subset, and by homology groups we mean singular homology groups over a finite field. However, in algebraic topology, there exists many other topological invariants than homology groups. In this talk, we will deal with Stiefel-Whitney classes. These classes are associated to any topological space endowed with a vector bundle structure, and they carry more topological information than the cohomology groups alone. Our new estimation problem is the following: given a finite sample of a vector bundle, estimate the Stiefel-Whitney classes of the vector bundle. We will adopt a persistent approach.

B. Wang: We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f: X → R. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (2018), we first show that the mapper graph of (X,f) approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (2017), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. We introduce a variant of the classic mapper graph of Singh et al. (2007), referred to as the enhanced mapper graph. We show that the enhanced mapper graph reduces the information loss during summarization and may be of independent interest. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data. This is joint work with Adam Brown,·Omer Bobrowski, and Elizabeth Munch.
https://arxiv.org/abs/1909.03488