{"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/tropical-optimal-transport-and-wasserstein","title":"Tropical Optimal Transport and Wasserstein Distances","arxiv_id":"1911.05401","date":"2019-11-13","proceeding":null,"authors":["Wonjun Lee","Wuchen Li","Bo Lin","Anthea Monod"],"abstract":"We study the problem of optimal transport in tropical geometry and define the Wasserstein-$p$ distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric -- a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees -- as the ground metric and study the cases of $p=1,2$ in detail. The case of $p=1$ gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of $p=2$ gives a form for Fr\\'{e}chet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport in tropical geometry. Several numerical examples are provided.","url_abs":"https://arxiv.org/abs/1911.05401v4","url_pdf":"https://arxiv.org/pdf/1911.05401v4.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":"tropical-optimal-transport-and-wasserstein","repo_url":"https://github.com/antheamonod/TropicalOT","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}