### Applying stratified homotopy theory in TDA

## Abstract

The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) on singular data sets. However, unlike in the non-singular scenario, the homotopy type (and consequently homology) are rather course invariants of singular spaces, even in low dimension. This suggests the use of finer invariants of singular spaces for TDA, making use of stratified homotopy theory instead of classical homotopy theory.

After an introduction to stratified homotopy theory, I will describe the construction of a persistent stratified homotopy type obtained from a sample with two strata. This construction behaves much like its non-stratified counterpart (the Cech complex) and exhibits many properties (such as stability, and inference results) necessary for an application in TDA.

Since the persistent stratified homotopy type relies on an already stratified point-cloud, I will also discuss the question of stratification learning and present a convergence result which allows one to approximately recover the stratifications of a larger class of two-strata stratified spaces from sufficiently close non-stratified samples. In total, these results combine to a sampling theorem guaranteeing the (approximate) inference of (persistent) stratified homotopy types from non-stratified samples for many examples of stratified spaces arising from geometrical scenarios.