# Metric reconstruction via optimal transport

## TL;DR

This paper introduces the Vietoris-Rips thickening, a metric space construction using optimal transport, which better captures the metric properties of a space and satisfies Hausmann's theorem for certain manifolds.

## Contribution

It defines the Vietoris-Rips thickening via optimal transport, providing a metrizable space that recovers the original space's homotopy type with a simpler proof for manifolds.

## Key findings

- Vietoris-Rips thickening satisfies Hausmann's theorem for complete Riemannian manifolds and compact Hadamard spaces.
- Homotopy equivalence is given by a canonical center of mass map.
- Describes the homotopy type of the thickening of the n-sphere at the first critical scale.

## Abstract

Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this approach is that $\mathrm{VR}(X;r)$ is not metrizable if it is not locally finite, and thus does not recover metric information about $M$. We attempt to remedy this shortcoming by defining a metric space thickening of $X$, which we call the \emph{Vietoris-Rips thickening} $\mathrm{VR}^m(X;r)$, via the theory of optimal transport. When $M$ is a complete Riemannian manifold, or alternatively a compact Hadamard space, we show that the the Vietoris-Rips thickening satisfies Hausmann's theorem ($\mathrm{VR}^m(M;r)\simeq M$ for $r$ sufficiently small) with a simpler proof: homotopy equivalence $\mathrm{VR}^m(M;r)\to M$ is canonically defined as a center of mass map, and its homotopy inverse is the (now continuous) inclusion map $M\hookrightarrow\mathrm{VR}^m(M;r)$. Furthermore, we describe the homotopy type of the Vietoris-Rips thickening of the $n$-sphere at the first positive scale parameter $r$ where the homotopy type changes.