{"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/fast-computation-of-wasserstein-barycenters","title":"Fast Computation of Wasserstein Barycenters","arxiv_id":"1310.4375","date":"2013-10-16","proceeding":null,"authors":["Marco Cuturi","Arnaud Doucet"],"abstract":"We present new algorithms to compute the mean of a set of empirical\nprobability measures under the optimal transport metric. This mean, known as\nthe Wasserstein barycenter, is the measure that minimizes the sum of its\nWasserstein distances to each element in that set. We propose two original\nalgorithms to compute Wasserstein barycenters that build upon the subgradient\nmethod. A direct implementation of these algorithms is, however, too costly\nbecause it would require the repeated resolution of large primal and dual\noptimal transport problems to compute subgradients. Extending the work of\nCuturi (2013), we propose to smooth the Wasserstein distance used in the\ndefinition of Wasserstein barycenters with an entropic regularizer and recover\nin doing so a strictly convex objective whose gradients can be computed for a\nconsiderably cheaper computational cost using matrix scaling algorithms. We use\nthese algorithms to visualize a large family of images and to solve a\nconstrained clustering problem.","url_abs":"http://arxiv.org/abs/1310.4375v3","url_pdf":"http://arxiv.org/pdf/1310.4375v3.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":"fast-computation-of-wasserstein-barycenters","repo_url":"https://github.com/nicolasbolle/barycenters","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":null},{"paper_slug":"fast-computation-of-wasserstein-barycenters","repo_url":"https://github.com/dan-mim/Wasserstein-barycenters/tree/main/Cuturi%20support%20and%20probability%20optimization","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"},{"task_slug":"constrained-clustering","task_name":"Constrained Clustering"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=1310.4375","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}