Function-Rips complexes in persistent homotopy theory: Local stability and Latschev theorems

TL;DR

This paper extends Latschev's theorem to persistent homotopy theory, establishing conditions under which the persistent homotopy type of a sublevel set filtration is stable under Gromov-Hausdorff perturbations, aiding in data analysis.

Contribution

It introduces a persistent version of Latschev's theorem, linking sublevel set filtrations and function-Rips complexes for nearby metric spaces and functions.

Findings

Provides conditions for persistent homotopy type stability.

Answers a longstanding question on estimating sublevel set persistent homology.

Establishes interleaving of persistent homotopy types under perturbations.

Abstract

Latschev's theorem provides sufficient conditions on a metric space $M$ and $\delta > 0$ for the homotopy type of $M$ to agree with that of the Vietoris-Rips complex $\mathcal{R}^{\delta}(N)$ of any nearby space $N$ in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair $(M, f \colon M \to \mathbb{R}^N)$ and $\delta > 0$ for the persistent homotopy type of the sublevel set filtration of $(M,f)$ to be interleaved with that of the function-Rips complex $\mathcal{R}^{\delta}(N^{\bullet})$ of any nearby pair $(N,g)$. In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.