{"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-concentration-of-the-empirical-measure-for","title":"On concentration of the empirical measure for radial transport costs","arxiv_id":"2305.18636","date":"2023-05-29","proceeding":null,"authors":["Martin Larsson","Jonghwa Park","Johannes Wiesel"],"abstract":"Let $\\mu$ be a probability measure on $\\mathbb{R}^d$ and $\\mu_N$ its empirical measure with sample size $N$. We prove a concentration inequality for the optimal transport cost between $\\mu$ and $\\mu_N$ for radial cost functions with polynomial local growth, that can have superpolynomial global growth. This result generalizes and improves upon estimates of Fournier and Guillin. The proof combines ideas from empirical process theory with known concentration rates for compactly supported $\\mu$. By partitioning $\\mathbb{R}^d$ into annuli, we infer a global estimate from local estimates on the annuli and conclude that the global estimate can be expressed as a sum of the local estimate and a mean-deviation probability for which efficient bounds are known.","url_abs":"https://arxiv.org/abs/2305.18636v2","url_pdf":"https://arxiv.org/pdf/2305.18636v2.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":"on-concentration-of-the-empirical-measure-for","repo_url":"https://github.com/jonghwap/concentration2023","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}