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The persistent topology of optimal transport based metric thickenings

Henry Adams, Facundo Mémoli, Michael Moy and Qingsong Wang

Algebraic & Geometric Topology 24 (2024) 393–447

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.

metric thickening, persistent homology, optimal transport, Fréchet variance
Mathematical Subject Classification
Primary: 55N31
Secondary: 51F99, 53C23
Received: 7 October 2021
Revised: 1 May 2022
Accepted: 1 August 2022
Published: 18 March 2024
Henry Adams
Department of Mathematics
University of Florida
Gainesville, FL
United States
Facundo Mémoli
Department of Mathematics and Department of Computer Science and Engineering
The Ohio State University
Columbus, OH
United States
Michael Moy
Department of Mathematics
Colorado State University
Fort Collins, CO
United States
Qingsong Wang
Department of Mathematics
The Ohio State University
Columbus, OH
United States

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