Wasserstein convergence of Cech persistence diagrams for samplings of submanifolds

Published: 25 Sept 2024, Last Modified: 06 Nov 2024NeurIPS 2024 posterEveryoneRevisionsBibTeXCC BY 4.0
Keywords: topological data analysis; persistence diagrams; simplicial complexes; differential geometry; Wasserstein distance
TL;DR: We study the convergence of Cech persistence diagrams for samples of submanifolds with respect to the Wasserstein distance.
Abstract:

Cech Persistence diagrams (PDs) are topological descriptors routinely used to capture the geometry of complex datasets. They are commonly compared using the Wasserstein distances $\mathrm{OT}_p$; however, the extent to which PDs are stable with respect to these metrics remains poorly understood. We partially close this gap by focusing on the case where datasets are sampled on an $m$-dimensional submanifold of $\mathbb{R}^d$. Under this manifold hypothesis, we show that convergence with respect to the $\mathrm{OT}_p$ metric happens exactly when $p>m$. We also provide improvements upon the bottleneck stability theorem in this case and prove new laws of large numbers for the total $\alpha$-persistence of PDs. Finally, we show how these theoretical findings shed new light on the behavior of the feature maps on the space of PDs that are used in ML-oriented applications of Topological Data Analysis.

Primary Area: Learning theory
Submission Number: 11246
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