{"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/sinkhorn-divergences-for-unbalanced-optimal","title":"Sinkhorn Divergences for Unbalanced Optimal Transport","arxiv_id":"1910.12958","date":"2019-10-28","proceeding":null,"authors":["Thibault Séjourné","Jean Feydy","François-Xavier Vialard","Alain Trouvé","Gabriel Peyré"],"abstract":"Optimal transport induces the Earth Mover's (Wasserstein) distance between probability distributions, a geometric divergence that is relevant to a wide range of problems. Over the last decade, two relaxations of optimal transport have been studied in depth: unbalanced transport, which is robust to the presence of outliers and can be used when distributions don't have the same total mass; entropy-regularized transport, which is robust to sampling noise and lends itself to fast computations using the Sinkhorn algorithm. This paper combines both lines of work to put robust optimal transport on solid ground. Our main contribution is a generalization of the Sinkhorn algorithm to unbalanced transport: our method alternates between the standard Sinkhorn updates and the pointwise application of a contractive function. This implies that entropic transport solvers on grid images, point clouds and sampled distributions can all be modified easily to support unbalanced transport, with a proof of linear convergence that holds in all settings. We then show how to use this method to define pseudo-distances on the full space of positive measures that satisfy key geometric axioms: (unbalanced) Sinkhorn divergences are differentiable, positive, definite, convex, statistically robust and avoid any \"entropic bias\" towards a shrinkage of the measures' supports.","url_abs":"https://arxiv.org/abs/1910.12958v3","url_pdf":"https://arxiv.org/pdf/1910.12958v3.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":"sinkhorn-divergences-for-unbalanced-optimal","repo_url":"https://github.com/thibsej/unbalanced-ot-functionals","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"sinkhorn-divergences-for-unbalanced-optimal","repo_url":"https://github.com/ericphanson/VisualStringDistances.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"sinkhorn-divergences-for-unbalanced-optimal","repo_url":"https://github.com/google-research/ott","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"jax","reach":{"status":"ok","spdx":"Apache-2.0"}},{"paper_slug":"sinkhorn-divergences-for-unbalanced-optimal","repo_url":"https://github.com/ott-jax/ott","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"jax","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1910.12958","atlas_url":"https://app.syntology.ai/?focus=1910.12958","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1910.12958"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/ott-jax/ott","reach":null},{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/ericphanson/VisualStringDistances.jl","reach":{"status":"ok","spdx":"NOASSERTION"}},{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/google-research/ott","reach":{"status":"ok","spdx":"Apache-2.0"}},{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/thibsej/unbalanced-ot-functionals","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"unverified":5},"by_repo_kind":{"official":{"samples":5,"ran":0,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":0,"samples":[{"code_sha256_prefix":"71c85d72e5e82243","entry":"dist_matrix","repo":"thibsej/unbalanced-ot-functionals","repo_kind":"official","path":"unbalancedot/utils.py","file_url":"https://github.com/thibsej/unbalanced-ot-functionals/blob/HEAD/unbalancedot/utils.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"71c85d72e5e82243"}},{"code_sha256_prefix":"966ef083aa6872f1","entry":"euclidean_cost","repo":"thibsej/unbalanced-ot-functionals","repo_kind":"official","path":"unbalancedot/utils.py","file_url":"https://github.com/thibsej/unbalanced-ot-functionals/blob/HEAD/unbalancedot/utils.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"966ef083aa6872f1"}},{"code_sha256_prefix":"811552d45071c8c2","entry":"init_lambertw","repo":"thibsej/unbalanced-ot-functionals","repo_kind":"official","path":"unbalancedot/torch_lambertw.py","file_url":"https://github.com/thibsej/unbalanced-ot-functionals/blob/HEAD/unbalancedot/torch_lambertw.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"811552d45071c8c2"}},{"code_sha256_prefix":"e4a1d524b992518f","entry":"log_lambertw","repo":"thibsej/unbalanced-ot-functionals","repo_kind":"official","path":"unbalancedot/torch_lambertw.py","file_url":"https://github.com/thibsej/unbalanced-ot-functionals/blob/HEAD/unbalancedot/torch_lambertw.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"e4a1d524b992518f"}},{"code_sha256_prefix":"a00500d4ed69d4b1","entry":"scal","repo":"thibsej/unbalanced-ot-functionals","repo_kind":"official","path":"unbalancedot/utils.py","file_url":"https://github.com/thibsej/unbalanced-ot-functionals/blob/HEAD/unbalancedot/utils.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"a00500d4ed69d4b1"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}