{"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/efficient-and-exact-multimarginal-optimal","title":"Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs","arxiv_id":"2208.03025","date":"2022-08-05","proceeding":null,"authors":["Bohan Zhou","Matthew Parno"],"abstract":"In this paper, we address the numerical solution to the multimarginal optimal transport (MMOT) with pairwise costs. MMOT, as a natural extension from the classical two-marginal optimal transport, has many important applications including image processing, density functional theory and machine learning, but yet lacks efficient and exact numerical methods. The popular entropy-regularized method may suffer numerical instability and blurring issues. Inspired by the back-and-forth method introduced by Jacobs and L\\'{e}ger, we investigate MMOT problems with pairwise costs. First, such problems have a graphical representation and we prove equivalent MMOT problems that have a tree representation. Second, we introduce a noval algorithm to solve MMOT on a rooted tree, by gradient based method on the dual formulation. Last, we obtain accurate solutions which can be used for the regularization-free applications.","url_abs":"https://arxiv.org/abs/2208.03025v3","url_pdf":"https://arxiv.org/pdf/2208.03025v3.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":"efficient-and-exact-multimarginal-optimal","repo_url":"https://github.com/simda-muri/mmot","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}