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Abstract

Singular spaces have gained prominence in diverse applications, generating interest in investigating stratified structures using methods within the field of topological data analysis (TDA). While persistent homology is a fundamental concept in TDA, its limitations in distinguishing stratified spaces prompt the need for alternative invariants. To address this need, we introduce the concept of persistent stratified homotopy types. We establish a series of properties analogous to those of the ordinary persistent homotopy type, such as stability, computability, and inference, which are crucial for TDA applications. In tackling the challenge of detecting singularities in data, we present methods to approximate stratifications from non-stratified data, sampled in the vicinity of sufficiently well-behaved two-strata Whitney stratified spaces. Our findings enable the formulation of a sampling theorem for approximately inferring (persistent) stratified homotopy types of suitably well-behaved two strata Whitney stratified spaces. Bridging theory and practical implementation, we introduce two distinct approaches to measure singularity in data. Leveraging insights from local persistent homology studies and employing the Hausdorff distance serve this purpose. Moreover, we delve into the topology of image patch spaces, providing novel insights and reassessing existing models, especially with regard to potential stratified structures.