The persistent topology of optimal transport based metric thickenings

Abstract

A metric thickening of a given metric space X is any metric space admitting an isometric embedding of X. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the p–Vietoris–Rips and p–Čech metric thickenings for all 1 ≤ p ≥ ∞, which include all probability measures on X whose p–diameter or p–radius is bounded from above, equipped with an optimal transport metric. The p–diameter (resp. p–radius) of a measure is a certain p relaxation of the usual notion of diameter (resp. radius) of a subset of a metric space. These families recover the previously studied Vietoris–Rips and Čech metric thickenings when p = ∞. As our main contribution, we prove a stability theorem for the persistent homology of p–Vietoris–Rips and p–Čech metric thickenings, which is novel even in the case p = ∞. In the specific case p = 2, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the 2–Vietoris–Rips thickenings of the n–sphere as the scale increases.