Quantitative stability of barycenters in the Wasserstein space

• 成果类型: Article
• 署名作者: Carlier, Guillaume; Delalande, Alex; Merigot, Quentin
• 署名单位: Universite PSL; Universite Paris-Dauphine; Inria; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; Institut Universitaire de France
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01241-5
• 发表日期: 2024
• 页码: 1257-1286
• 关键词: Transportation; CONVERGENCE; convexity

摘要

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 Holder-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.