{"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/comparing-distributions-ell_1-geometry-1","title":"Comparing distributions: \\ell_1 geometry improves kernel two-sample testing","arxiv_id":null,"date":"2019-12-01","proceeding":"NeurIPS 2019 12","authors":["Meyer Scetbon","Gael Varoquaux"],"abstract":"Are two sets of observations drawn from the same distribution? This\nproblem is a two-sample test. \nKernel methods lead to many appealing properties. Indeed state-of-the-art\napproaches use the $L^2$ distance between kernel-based\ndistribution representatives to derive their test statistics. Here, we show that\n$L^p$ distances (with $p\\geq 1$) between these\ndistribution representatives give metrics on the space of distributions that are\nwell-behaved to detect differences between distributions as they\nmetrize the weak convergence. Moreover, for analytic kernels,\nwe show that the $L^1$ geometry gives improved testing power for\nscalable computational procedures. Specifically, we derive a finite\ndimensional approximation of the metric given as the $\\ell_1$ norm of a vector which captures differences of expectations of analytic functions evaluated at spatial locations or frequencies (i.e, features). The features can be chosen to\nmaximize the differences of the distributions and give interpretable\nindications of how they differs. Using an $\\ell_1$ norm gives better detection\nbecause differences between representatives are dense\nas we use analytic kernels (non-zero almost everywhere). The tests are consistent, while\nmuch faster than state-of-the-art quadratic-time kernel-based tests. Experiments\non artificial\nand real-world problems demonstrate\nimproved power/time tradeoff than the state of the art, based on\n$\\ell_2$ norms, and in some cases, better outright power than even the most\nexpensive quadratic-time tests. This performance gain is retained even in high dimensions.","url_abs":"http://papers.nips.cc/paper/9398-comparing-distributions-ell_1-geometry-improves-kernel-two-sample-testing","url_pdf":"http://papers.nips.cc/paper/9398-comparing-distributions-ell_1-geometry-improves-kernel-two-sample-testing.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":"comparing-distributions-ell_1-geometry-1","repo_url":"https://github.com/meyerscetbon/l1_two_sample_test","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"hypothesis-testing","task_name":"Two-sample testing"},{"task_slug":"two","task_name":"Vocal Bursts Valence Prediction"}],"methods":[{"method_slug":"test","method_name":"Test"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}