Lukas Waas: From Samples to Persistent Stratified Homotopy Types

Abstract: 

The natural occurrence of stratified spaces in application has led to recent investigations on how to perform topological data analysis (TDA) in a stratified framework. Non-stratified applications in TDA rely on a series of convenient properties of (persistent) homotopy types of sufficiently regular spaces. Classical notions of stratified homotopy equivalence generally turn out to be too rigid to display similar behavior. At the same time, recent developments in the field of abstract stratified homotopy theory employ weaker notions of stratified equivalences. These exhibit behavior more suitable for the purpose of TDA, while still retaining the same homotopy theoretical information as classical stratified homotopy equivalences, at least for classical examples such as Whitney stratified spaces.

In this work, we establish a notion of persistent stratified homotopy type, obtained from a stratified sample. We show that it behaves much like its non-stratified counterpart and exhibits many of the properties (such as stability) expected for an application in TDA. Using local approximations of tangent cones, we then describe a pipeline which constructs a persistent stratified homotopy type from a non-stratified sample, i.e. without any prior knowledge of the location of the singularity. Our main result is a sampling theorem, guaranteeing that for certain Whitney stratified spaces our method does indeed produce arbitrarily close approximations of the persistent stratified homotopy type of the original space, as long as sufficiently good samples are provided.