{"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/computational-optimal-transport-complexity-by-1","title":"Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm","arxiv_id":"1802.04367","date":"2018-02-12","proceeding":null,"authors":["Pavel Dvurechensky","Alexander Gasnikov","Alexey Kroshnin"],"abstract":"We analyze two algorithms for approximating the general optimal transport (OT) distance between two discrete distributions of size $n$, up to accuracy $\\varepsilon$. For the first algorithm, which is based on the celebrated Sinkhorn's algorithm, we prove the complexity bound $\\widetilde{O}\\left({n^2/\\varepsilon^2}\\right)$ arithmetic operations. For the second one, which is based on our novel Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD) algorithm, we prove the complexity bound $\\widetilde{O}\\left(\\min\\left\\{n^{9/4}/\\varepsilon, n^{2}/\\varepsilon^2 \\right\\}\\right)$ arithmetic operations. Both bounds have better dependence on $\\varepsilon$ than the state-of-the-art result given by $\\widetilde{O}\\left({n^2/\\varepsilon^3}\\right)$. Our second algorithm not only has better dependence on $\\varepsilon$ in the complexity bound, but also is not specific to entropic regularization and can solve the OT problem with different regularizers.","url_abs":"https://arxiv.org/abs/1802.04367v2","url_pdf":"https://arxiv.org/pdf/1802.04367v2.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":"computational-optimal-transport-complexity-by-1","repo_url":"https://github.com/kumarak93/numpy_ot","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1802.04367","atlas_url":"https://app.syntology.ai/?focus=1802.04367","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}