{"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/unbalanced-optimal-transport-through-non","title":"Unbalanced Optimal Transport through Non-negative Penalized Linear Regression","arxiv_id":"2106.04145","date":"2021-06-08","proceeding":"NeurIPS 2021 12","authors":["Laetitia Chapel","Rémi Flamary","Haoran Wu","Cédric Févotte","Gilles Gasso"],"abstract":"This paper addresses the problem of Unbalanced Optimal Transport (UOT) in which the marginal conditions are relaxed (using weighted penalties in lieu of equality) and no additional regularization is enforced on the OT plan. In this context, we show that the corresponding optimization problem can be reformulated as a non-negative penalized linear regression problem. This reformulation allows us to propose novel algorithms inspired from inverse problems and nonnegative matrix factorization. In particular, we consider majorization-minimization which leads in our setting to efficient multiplicative updates for a variety of penalties. Furthermore, we derive for the first time an efficient algorithm to compute the regularization path of UOT with quadratic penalties. The proposed algorithm provides a continuity of piece-wise linear OT plans converging to the solution of balanced OT (corresponding to infinite penalty weights). We perform several numerical experiments on simulated and real data illustrating the new algorithms, and provide a detailed discussion about more sophisticated optimization tools that can further be used to solve OT problems thanks to our reformulation.","url_abs":"https://arxiv.org/abs/2106.04145v1","url_pdf":"https://arxiv.org/pdf/2106.04145v1.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":"unbalanced-optimal-transport-through-non","repo_url":"https://github.com/lchapel/UOT-though-penalized-linear-regression","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"regression-1","task_name":"regression"}],"methods":[{"method_slug":"linear-regression","method_name":"Linear Regression"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2106.04145","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2106.04145"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+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/lchapel/UOT-though-penalized-linear-regression","reach":null}],"summary":{"ran_honours":2,"ran_fixture":2,"ran_draft_wrong":2},"by_repo_kind":{"listed":{"samples":6,"ran":6,"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":6,"samples":[{"code_sha256_prefix":"a2ce9978525740bf","entry":"KL_divergence","repo":"lchapel/UOT-though-penalized-linear-regression","repo_kind":"listed","path":"solvers/solver_kl_UOT.py","file_url":"https://github.com/lchapel/UOT-though-penalized-linear-regression/blob/HEAD/solvers/solver_kl_UOT.py","link_basis":"first_harvest_node","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"a2ce9978525740bf"}},{"code_sha256_prefix":"dd86526c733d446c","entry":"complement_schur","repo":"lchapel/UOT-though-penalized-linear-regression","repo_kind":"listed","path":"solvers/solvers_L2_UOT.py","file_url":"https://github.com/lchapel/UOT-though-penalized-linear-regression/blob/HEAD/solvers/solvers_L2_UOT.py","link_basis":"first_harvest_node","language":"python","status":"ran_fixture","verification_level":1,"contract_check":"RAISES","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"dd86526c733d446c"}},{"code_sha256_prefix":"5dfe5360f0d13263","entry":"compute_lambda_a","repo":"lchapel/UOT-though-penalized-linear-regression","repo_kind":"listed","path":"solvers/solvers_L2_UOT.py","file_url":"https://github.com/lchapel/UOT-though-penalized-linear-regression/blob/HEAD/solvers/solvers_L2_UOT.py","link_basis":"first_harvest_node","language":"python","status":"ran_fixture","verification_level":1,"contract_check":"RAISES","metamorphic_tier":"invariant","behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"5dfe5360f0d13263"}},{"code_sha256_prefix":"0ed106a410e4b3d9","entry":"compute_lambda_r","repo":"lchapel/UOT-though-penalized-linear-regression","repo_kind":"listed","path":"solvers/solvers_L2_UOT.py","file_url":"https://github.com/lchapel/UOT-though-penalized-linear-regression/blob/HEAD/solvers/solvers_L2_UOT.py","link_basis":"first_harvest_node","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"0ed106a410e4b3d9"}},{"code_sha256_prefix":"0df8bde447cf4521","entry":"get_X_lasso","repo":"lchapel/UOT-though-penalized-linear-regression","repo_kind":"listed","path":"solvers/solver_kl_UOT.py","file_url":"https://github.com/lchapel/UOT-though-penalized-linear-regression/blob/HEAD/solvers/solver_kl_UOT.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"OUTPUT_MISDECLARED","metamorphic_tier":"deterministic","behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"0df8bde447cf4521"}},{"code_sha256_prefix":"545e31db79e8780f","entry":"ot_ukl_solve_mu","repo":"lchapel/UOT-though-penalized-linear-regression","repo_kind":"listed","path":"solvers/solver_kl_UOT.py","file_url":"https://github.com/lchapel/UOT-though-penalized-linear-regression/blob/HEAD/solvers/solver_kl_UOT.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"MISDECLARED","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"545e31db79e8780f"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}