Multidimensional Interleavings and Applications to Topological Inference

TL;DR

This paper develops a theoretical framework for multidimensional persistence in topological data analysis, introducing generalized interleavings and interleaving distances to improve topological inference directly at the filtration level.

Contribution

It generalizes the concept of interleavings to multidimensional filtrations and modules, enabling direct topological inference at the filtration level.

Findings

Defined new pseudometrics on multidimensional filtrations

Proved multidimensional analogues of topological inference theorems

Established foundations for direct topological inference in multidimensional persistence

Abstract

This work concerns the theoretical foundations of persistence-based topological data analysis. We develop theory of topological inference in the multidimensional persistence setting, and directly at the (topological) level of filtrations rather than only at the (algebraic) level of persistent homology modules. Our main mathematical objects of study are interleavings. These are tools for quantifying the similarity between two multidimensional filtrations or persistence modules. They were introduced for 1-D filtrations and persistence modules by Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot. We introduce generalizations of the definitions of interleavings given by Chazal et al. and use these to define pseudometrics, called interleaving distances, on multidimensional filtrations and multidimensional persistence modules. We present an in-depth study of interleavings and interleaving distances. We then use them to formulate and prove several multidimensional analogues of a topological inference theorem of Chazal, Guibas, Oudot, and Skraba. These results hold directly at the level of filtrations; they yield as corollaries corresponding results at the module level.