Quantitative stability of barycenters in the Wasserstein space

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 enti...

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Bibliographic Details
Published in:Probability theory and related fields 2024-04, Vol.188 (3-4), p.1257-1286
Main Authors: Carlier, Guillaume, Delalande, Alex, Mérigot, Quentin
Format: Article
Language:English
Subjects:
Economics
Finance
Insurance
Management
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Operations Research/Decision Theory
Optimization and Control
Probability Theory and Stochastic Processes
Quantitative Finance
Statistics
Statistics for Business
Theoretical
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Summary: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 allow to quantify the strong convexity of the barycenter functional. Consequences regarding the statistical estimation of Wasserstein barycenters and the convergence of regularized Wasserstein barycenters towards their non-regularized counterparts are explored.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-023-01241-5