{"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/on-the-complexity-of-approximating","title":"On the Complexity of Approximating Multimarginal Optimal Transport","arxiv_id":"1910.00152","date":"2019-09-30","proceeding":null,"authors":["Tianyi Lin","Nhat Ho","Marco Cuturi","Michael. I. Jordan"],"abstract":"We study the complexity of approximating the multimarginal optimal transport (MOT) distance, a generalization of the classical optimal transport distance, considered here between $m$ discrete probability distributions supported each on $n$ support points. First, we show that the standard linear programming (LP) representation of the MOT problem is not a minimum-cost flow problem when $m \\geq 3$. This negative result implies that some combinatorial algorithms, e.g., network simplex method, are not suitable for approximating the MOT problem, while the worst-case complexity bound for the deterministic interior-point algorithm remains a quantity of $\\tilde{O}(n^{3m})$. We then propose two simple and \\textit{deterministic} algorithms for approximating the MOT problem. The first algorithm, which we refer to as \\textit{multimarginal Sinkhorn} algorithm, is a provably efficient multimarginal generalization of the Sinkhorn algorithm. We show that it achieves a complexity bound of $\\tilde{O}(m^3n^m\\varepsilon^{-2})$ for a tolerance $\\varepsilon \\in (0, 1)$. This provides a first \\textit{near-linear time} complexity bound guarantee for approximating the MOT problem and matches the best known complexity bound for the Sinkhorn algorithm in the classical OT setting when $m = 2$. The second algorithm, which we refer to as \\textit{accelerated multimarginal Sinkhorn} algorithm, achieves the acceleration by incorporating an estimate sequence and the complexity bound is $\\tilde{O}(m^3n^{m+1/3}\\varepsilon^{-4/3})$. This bound is better than that of the first algorithm in terms of $1/\\varepsilon$, and accelerated alternating minimization algorithm~\\citep{Tupitsa-2020-Multimarginal} in terms of $n$. Finally, we compare our new algorithms with the commercial LP solver \\textsc{Gurobi}. Preliminary results on synthetic data and real images demonstrate the effectiveness and efficiency of our algorithms.","url_abs":"https://arxiv.org/abs/1910.00152v4","url_pdf":"https://arxiv.org/pdf/1910.00152v4.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":"on-the-complexity-of-approximating","repo_url":"https://github.com/shuge-mit/mot_project","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1910.00152","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1910.00152"}},"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. 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