Persistence stability for geometric complexes

TL;DR

This paper investigates the stability of persistent homology in geometric complexes like Vietoris-Rips and Cech, providing new proofs and properties related to their homology on precompact and compact spaces.

Contribution

It offers simple proofs of stability of persistent homology under Gromov-Hausdorff distance and explores properties of homology in complexes on compact spaces.

Findings

Persistent homology is stable under Gromov-Hausdorff distance.

Homology properties of Rips and Cech complexes on compact spaces.

New insights into the behavior of geometric complexes' homology.

Abstract

In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on top of precompact spaces. Using recent developments in the theory of topological persistence we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov--Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Cech complexes built on top of compact spaces.