{"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/sobolev-transport-a-scalable-metric-for","title":"Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics","arxiv_id":"2202.10723","date":"2022-02-22","proceeding":null,"authors":["Tam Le","Truyen Nguyen","Dinh Phung","Viet Anh Nguyen"],"abstract":"Optimal transport (OT) is a popular measure to compare probability distributions. However, OT suffers a few drawbacks such as (i) a high complexity for computation, (ii) indefiniteness which limits its applicability to kernel machines. In this work, we consider probability measures supported on a graph metric space and propose a novel Sobolev transport metric. We show that the Sobolev transport metric yields a closed-form formula for fast computation and it is negative definite. We show that the space of probability measures endowed with this transport distance is isometric to a bounded convex set in a Euclidean space with a weighted $\\ell_p$ distance. We further exploit the negative definiteness of the Sobolev transport to design positive-definite kernels, and evaluate their performances against other baselines in document classification with word embeddings and in topological data analysis.","url_abs":"https://arxiv.org/abs/2202.10723v1","url_pdf":"https://arxiv.org/pdf/2202.10723v1.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":"sobolev-transport-a-scalable-metric-for","repo_url":"https://github.com/lttam/sobolevtransport","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"document-classification","task_name":"Document Classification"},{"task_slug":"topological-data-analysis","task_name":"Topological Data Analysis"},{"task_slug":"word-embeddings","task_name":"Word Embeddings"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2202.10723","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}