Stratified Topological Data Analysis: Theory, Methods and Applications in Image Patch Spaces

Singular spaces are of growing interest across various applications, prompting exploration through topological data analysis (TDA). While persistent homology is a key tool, its limitations in distinguishing stratified spaces necessitate alternative invariants. To address this, we propose a novel framework with properties akin to persistent homotopy types, crucial for TDA applications. Furthermore, we present fully implementable methods that enable us to approximate stratifications of a large class of stratified spaces from non-stratified data. Additionally, we introduce two approaches to measure singularity, drawing on insights from local persistent homology and Hausdorff distance. Furthermore, we investigate image patch spaces, offering new perspectives and reevaluation of existing models, especially regarding potential stratified structures.