{"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/robust-first-and-second-order-differentiation","title":"Robust First and Second-Order Differentiation for Regularized Optimal Transport","arxiv_id":"2407.02015","date":"2024-07-02","proceeding":null,"authors":["Xingjie Li","Fei Lu","Molei Tao","Felix X. -F. Ye"],"abstract":"Applications such as unbalanced and fully shuffled regression can be approached by optimizing regularized optimal transport (OT) distances, such as the entropic OT and Sinkhorn distances. A common approach for this optimization is to use a first-order optimizer, which requires the gradient of the OT distance. For faster convergence, one might also resort to a second-order optimizer, which additionally requires the Hessian. The computations of these derivatives are crucial for efficient and accurate optimization. However, they present significant challenges in terms of memory consumption and numerical instability, especially for large datasets and small regularization strengths. We circumvent these issues by analytically computing the gradients for OT distances and the Hessian for the entropic OT distance, which was not previously used due to intricate tensor-wise calculations and the complex dependency on parameters within the bi-level loss function. Through analytical derivation and spectral analysis, we identify and resolve the numerical instability caused by the singularity and ill-posedness of a key linear system. Consequently, we achieve scalable and stable computation of the Hessian, enabling the implementation of the stochastic gradient descent (SGD)-Newton methods. Tests on shuffled regression examples demonstrate that the second stage of the SGD-Newton method converges orders of magnitude faster than the gradient descent-only method while achieving significantly more accurate parameter estimations.","url_abs":"https://arxiv.org/abs/2407.02015v2","url_pdf":"https://arxiv.org/pdf/2407.02015v2.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"robust-first-and-second-order-differentiation","repo_url":"https://github.com/yexf308/OTT-Hessian","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2407.02015","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2407.02015"}},"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/yexf308/OTT-Hessian","reach":{"status":"ok"}}],"summary":{"ran":3},"by_repo_kind":{"official":{"samples":3,"ran":3,"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":3,"samples":[{"code_sha256_prefix":"438b7ecc339b4506","entry":"sample_bolb","repo":"yexf308/OTT-Hessian","repo_kind":"official","path":"util.py","file_url":"https://github.com/yexf308/OTT-Hessian/blob/HEAD/util.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"438b7ecc339b4506"}},{"code_sha256_prefix":"776e0cd7dc09aed1","entry":"sample_points_gaussian","repo":"yexf308/OTT-Hessian","repo_kind":"official","path":"util.py","file_url":"https://github.com/yexf308/OTT-Hessian/blob/HEAD/util.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"776e0cd7dc09aed1"}},{"code_sha256_prefix":"6281160c3a57eb90","entry":"sample_points_uniform","repo":"yexf308/OTT-Hessian","repo_kind":"official","path":"util.py","file_url":"https://github.com/yexf308/OTT-Hessian/blob/HEAD/util.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"6281160c3a57eb90"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}