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The persistent topology
of optimal transport based metric thickenings
Henry Adams, Facundo Mémoli, Michael Moy and Qingsong Wang |
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Algebraic & Geometric Topology 24 (2024) 393–447
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Abstract |
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A metric thickening of a given metric space is any metric space admitting an isometric embedding of . 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 –Vietoris–Rips and –Čech metric thickenings for all , which include all probability measures on whose –diameter or –radius is bounded from above, equipped with an optimal transport metric. The –diameter (resp. –radius) of a measure is a certain 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 . As our main contribution, we prove a stability theorem for the persistent homology of –Vietoris–Rips and –Čech metric thickenings, which is novel even in the case . In the specific case , we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the –Vietoris–Rips thickenings of the –sphere as the scale increases. |
Keywords
metric thickening, persistent homology, optimal transport,
Fréchet variance
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Mathematical Subject Classification
Primary: 55N31
Secondary: 51F99, 53C23
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References |
Publication
Received: 7 October 2021
Revised: 1 May 2022
Accepted: 1 August 2022
Published: 18 March 2024
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Authors |
| Henry Adams | |
| Department of Mathematics University of Florida Gainesville, FL United States |
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| Facundo Mémoli | |
| Department of Mathematics and
Department of Computer Science and Engineering The Ohio State University Columbus, OH United States |
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| Michael Moy | |
| Department of Mathematics Colorado State University Fort Collins, CO United States |
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| Qingsong Wang | |
| Department of Mathematics The Ohio State University Columbus, OH United States |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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