Metric Reconstruction Via Optimal Transport

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