Euler and Betti curves are stable under Wasserstein deformations of distributions of stochastic processes

TL;DR

This paper proves that Euler and Betti curves, used in topological data analysis of stochastic processes on manifolds, are stable under Wasserstein perturbations of their distributions, ensuring robustness of these topological summaries.

Contribution

It establishes Wasserstein stability of Euler and Betti curves for stochastic processes on manifolds, extending the robustness guarantees in topological data analysis.

Findings

Euler and Betti curves are Wasserstein stable for processes in Sobolev spaces.

Stability holds for all p > d/n for persistence diagrams from Sobolev functions.

Provides theoretical guarantees for topological summaries under distribution perturbations.

Abstract

Euler and Betti curves of stochastic processes defined on a $d$-dimensional compact Riemannian manifold which are almost surely in a Sobolev space $W^{n,s}(X,\mathbb{R})$ (with $d<n$) are stable under perturbations of the distributions of said processes in a Wasserstein metric. Moreover, Wasserstein stability is shown to hold for all $p>\frac{d}{n}$ for persistence diagrams stemming from functions in $W^{n,s}(X,\mathbb{R})$.