From Samples to Persistent Stratified Homotopy Types

TL;DR

This paper develops a method to approximate and analyze the persistent stratified homotopy types of singular spaces from samples, enabling topological data analysis in stratified settings.

Contribution

It introduces a provably convergent process for stratification from samples and defines a persistent stratified homotopy type with stability properties.

Findings

Provides a sampling theorem for stratified homotopy types.

Establishes stability and inference properties for the persistent stratified homotopy type.

Demonstrates applicability to two-strata Whitney stratified spaces.

Abstract

The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) in a stratified framework. In many applications, there is no a priori information on what points should be regarded as singular or regular. For this purpose we describe a fully implementable process that provably approximates the stratification for a large class of two-strata Whitney stratified spaces from sufficiently close non-stratified samples. Additionally, in this work, we establish a notion of persistent stratified homotopy type obtained from a sample with two strata. In analogy to the non-stratified applications in TDA which rely on a series of convenient properties of (persistent) homotopy types of sufficiently regular spaces, we show that our persistent stratified homotopy type behaves much like its non-stratified counterpart and exhibits many properties (such as stability, and inference results) necessary for an application in TDA. In total, our results combine to a sampling theorem guaranteeing the (approximate) inference of (persistent) stratified homotopy types of sufficiently regular two-strata Whitney stratified spaces.