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Quantitative Stability of Barycenters in the Wasserstein Space

Abstract : Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability measures of interest are often not accessible in their entirety and the practitioner may have to deal with statistical or computational approximations instead. In this article, we quantify the effect of such approximations on the corresponding barycenters. We show that Wasserstein barycenters depend in a Hölder-continuous way on their marginals under relatively mild assumptions. Our proof relies on recent estimates that quantify the strong convexity of the dual quadratic optimal transport problem and a new result that allows to control the modulus of continuity of the push-forward operation under a (not necessarily smooth) optimal transport map.
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https://hal.archives-ouvertes.fr/hal-03781835
Contributor : Alex Delalande Connect in order to contact the contributor
Submitted on : Tuesday, September 20, 2022 - 4:58:16 PM
Last modification on : Thursday, November 3, 2022 - 9:07:07 AM

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  • HAL Id : hal-03781835, version 1
  • ARXIV : 2209.10217

Citation

Guillaume Carlier, Alex Delalande, Quentin Merigot. Quantitative Stability of Barycenters in the Wasserstein Space. 2022. ⟨hal-03781835⟩

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