Persistent Cost of Lipschitz Maps

TL;DR

This paper introduces the concept of persistent cost for Lipschitz maps between metric spaces, linking it to the stability of persistent homology and providing bounds based on metric properties.

Contribution

It defines persistent cost for Lipschitz maps, relates it to interleaving distance, and proves its stability using a Gromov-Hausdorff type metric.

Findings

Persistent cost controls interleaving distance between persistence modules.

Explicit upper bounds for persistent cost in metric terms.

Proof of stability of persistent cost under metric perturbations.

Abstract

A $1$-Lipschitz map between compact metric spaces $f\colon X\to Y$ induces a homomorphism of persistence modules on degree-$d$ Vietoris--Rips persistent homology. We define the persistent cost of $f$ from this induced homomorphism by quantifying the persistence carried by its kernel and cokernel modules. We prove that the persistent cost controls the interleaving distance between the degree-$d$ Vietoris--Rips persistent homology modules of $X$ and $Y$. Moreover, we obtain an explicit upper bound for the persistent cost in purely metric terms. Finally, we give a self-contained proof of the stability of the persistent cost introducing a Gromov-Hausdorff type distance for maps between compact metric spaces.