{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/doubly-regularized-entropic-wasserstein","title":"Doubly Regularized Entropic Wasserstein Barycenters","arxiv_id":"2303.11844","date":"2023-03-21","proceeding":null,"authors":["Lénaïc Chizat"],"abstract":"We study a general formulation of regularized Wasserstein barycenters that enjoys favorable regularity, approximation, stability and (grid-free) optimization properties. This barycenter is defined as the unique probability measure that minimizes the sum of entropic optimal transport (EOT) costs with respect to a family of given probability measures, plus an entropy term. We denote it $(\\lambda,\\tau)$-barycenter, where $\\lambda$ is the inner regularization strength and $\\tau$ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of $\\lambda,\\tau \\geq 0$ and generalizes them. First, in spite of -- and in fact owing to -- being \\emph{doubly} regularized, we show that our formulation is debiased for $\\tau=\\lambda/2$: the suboptimality in the (unregularized) Wasserstein barycenter objective is, for smooth densities, of the order of the strength $\\lambda^2$ of entropic regularization, instead of $\\max\\{\\lambda,\\tau\\}$ in general. We discuss this phenomenon for isotropic Gaussians where all $(\\lambda,\\tau)$-barycenters have closed form. Second, we show that for $\\lambda,\\tau>0$, this barycenter has a smooth density and is strongly stable under perturbation of the marginals. In particular, it can be estimated efficiently: given $n$ samples from each of the probability measures, it converges in relative entropy to the population barycenter at a rate $n^{-1/2}$. And finally, this formulation lends itself naturally to a grid-free optimization algorithm: we propose a simple \\emph{noisy particle gradient descent} which, in the mean-field limit, converges globally at an exponential rate to the barycenter.","url_abs":"https://arxiv.org/abs/2303.11844v1","url_pdf":"https://arxiv.org/pdf/2303.11844v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"doubly-regularized-entropic-wasserstein","repo_url":"https://github.com/lchizat/2023-doubly-entropic-barycenter","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2303.11844","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}