{"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/the-ultrametric-gromov-wasserstein-distance","title":"The ultrametric Gromov-Wasserstein distance","arxiv_id":"2101.05756","date":"2021-01-14","proceeding":null,"authors":["Facundo Mémoli","Axel Munk","Zhengchao Wan","Christoph Weitkamp"],"abstract":"In this paper, we investigate compact ultrametric measure spaces which form a subset $\\mathcal{U}^w$ of the collection of all metric measure spaces $\\mathcal{M}^w$. Similar as for the ultrametric Gromov-Hausdorff distance on the collection of ultrametric spaces $\\mathcal{U}$, we define ultrametric versions of two metrics on $\\mathcal{U}^w$, namely of Sturm's distance of order $p$ and of the Gromov-Wasserstein distance of order $p$. We study the basic topological and geometric properties of these distances as well as their relation and derive for $p=\\infty$ a polynomial time algorithm for their calculation. Further, several lower bounds for both distances are derived and some of our results are generalized to the case of finite ultra-dissimilarity spaces.","url_abs":"https://arxiv.org/abs/2101.05756v2","url_pdf":"https://arxiv.org/pdf/2101.05756v2.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":"the-ultrametric-gromov-wasserstein-distance","repo_url":"https://github.com/ndag/uGW","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}