Estimating Multidimensional Persistent Homology through a Finite Sampling

TL;DR

This paper demonstrates that under certain density conditions, the multidimensional persistent Betti numbers of a submanifold can be estimated from finite samples using unions of balls, improving shape dissimilarity measures.

Contribution

It introduces a method to estimate multidimensional persistent Betti numbers from finite samples, providing exact values in certain regions and enhancing shape comparison metrics.

Findings

Estimation of Betti numbers from finite samples is feasible under density conditions.

Improved lower bounds for the pseudodistance between shapes are achieved.

Inequalities are established for Betti numbers of ball unions and their combinatorial models.

Abstract

An exact computation of the persistent Betti numbers of a submanifold $X$ of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of $X$ is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of $X$ from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.