Henry Adams

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  Metric reconstruction via optimal transport
  Henry Adams
  Given a sample of points X from a metric space M, the Vietoris-Rips simplicial complex VR(X;r) at scale r>0 is a standard construction to attempt to recover M from X, up to homotopy type. A deficiency is that VR(X;r) is not metrizable if it is not locally finite, and thus does not recover metric information about M. We remedy this shortcoming by defining the Vietoris-Rips metric thickening VR^m(X;r) via the theory of optimal transport. Vertices are reinterpreted as Dirac delta masses, points in simplices are reinterpreted as convex combinations of Dirac delta masses, and distances are given by the Wasserstein distance between probability measures. When M is a Riemannian manifold, the Vietoris-Rips thickening satisfies Hausmann's theorem (VR^m(M;r) is homotopy equivalent to M for r sufficiently small) with a simpler proof: homotopy equivalence VR^m(M;r) -> M is now canonically defined as a center of mass map, and its homotopy inverse is the (now continuous) inclusion M -> VR^m(M;r). We discuss Vietoris-Rips thickenings of circles and n-spheres, and relate these constructions to Borsuk-Ulam theorems into higher-dimensional codomains. Joint work with Michal Adamaszek, John Bush, and Florian Frick.