Metric Reconstruction Via Optimal Transport

Abstract

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 VRm(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 (VRm(X; r) \simeq M for r sufficiently small) with a simpler proof than Hausmann's original result: homotopy equivalence VRm(X; r) \rightarrow M is canonically defined as a center of mass map, and its homotopy inverse is the (now continuous) inclusion map M \lhook \rightarrow VRm(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.