Metric thickenings of Euclidean submanifolds

TL;DR

This paper introduces metric thickenings of Euclidean submanifolds using the 1-Wasserstein metric, establishing homotopy equivalences with the original shape for sets with positive reach, extending classical topological results.

Contribution

It proves a metric version of Hausmann's theorem for Euclidean submanifolds and non-manifold shapes using Wasserstein metric thickenings, with explicit deformation retraction homotopies.

Findings

Metric thickenings are homotopy equivalent to the original set for scales less than the reach.

Homotopy equivalence is realized by canonical maps and simple linear homotopies.

First such result for Euclidean submanifolds extending classical topological theorems.

Abstract

Given a sample $Y$ from an unknown manifold $X$ embedded in Euclidean space, it is possible to recover the homology groups of $X$ by building a Vietoris--Rips or \v{C}ech simplicial complex on top of the vertex set $Y$. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular when $Y$ is infinite. Indeed, a simplicial complex is not even metrizable if it is not locally finite. We instead consider metric thickenings, called the \emph{Vietoris--Rips} and \emph{\v{C}ech thickenings}, which are equipped with the 1-Wasserstein metric in place of the simplicial complex topology. We show that for Euclidean subsets $X$ with positive reach, the thickenings satisfy metric analogues of Hausmann's theorem and the nerve lemma (the metric Vietoris--Rips and \v{C}ech thickenings of $X$ are homotopy equivalent to $X$ for scale parameters less than the reach). To our knowledge this is the first version of Hausmann's theorem for Euclidean submanifolds (as opposed to Riemannian manifolds), and our result also extends to non-manifold shapes (as not all sets of positive reach are manifolds). In contrast to Hausmann's original proof, our homotopy equivalence is a deformation retraction, is realized by canonical maps in both directions, and furthermore can be proven to be a homotopy equivalence via simple linear homotopies from the map compositions to the corresponding identity maps.