Wasserstein Stability for Persistence Diagrams

TL;DR

This paper introduces new stability results for persistence diagrams using the p-Wasserstein distance, offering more robust bounds than traditional bottleneck distance, with applications in topological data analysis.

Contribution

It provides novel stability theorems for persistence diagrams under p-Wasserstein distance, extending existing results and including an algebraic framework for broader applicability.

Findings

New stability bounds for persistence diagrams using p-Wasserstein distance

Elementary proof for finite spaces with p-norm perturbations

Applications to topological summaries and Vietoris-Rips complexes

Abstract

The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of perturbations. This has two main implications: it makes the space of persistence diagrams rather pathological and it is often provides very pessimistic bounds with respect to outliers. In this paper, we provide new stability results with respect to the $p$-Wasserstein distance between persistence diagrams. This includes an elementary proof for the setting of functions on sufficiently finite spaces in terms of the $p$-norm of the perturbations, along with an algebraic framework for $p$-Wasserstein distance which extends the results to wider class of modules. We also provide apply the results to a wide range of applications in topological data analysis (TDA) including topological summaries, persistence transforms and the special but important case of Vietoris-Rips complexes.