{"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/kernel-mmd-two-sample-tests-for-manifold-data","title":"Kernel Two-Sample Tests for Manifold Data","arxiv_id":"2105.03425","date":"2021-05-07","proceeding":null,"authors":["Xiuyuan Cheng","Yao Xie"],"abstract":"We present a study of a kernel-based two-sample test statistic related to the Maximum Mean Discrepancy (MMD) in the manifold data setting, assuming that high-dimensional observations are close to a low-dimensional manifold. We characterize the test level and power in relation to the kernel bandwidth, the number of samples, and the intrinsic dimensionality of the manifold. Specifically, when data densities $p$ and $q$ are supported on a $d$-dimensional sub-manifold ${M}$ embedded in an $m$-dimensional space and are H\\\"older with order $\\beta$ (up to 2) on ${M}$, we prove a guarantee of the test power for finite sample size $n$ that exceeds a threshold depending on $d$, $\\beta$, and $\\Delta_2$ the squared $L^2$-divergence between $p$ and $q$ on the manifold, and with a properly chosen kernel bandwidth $\\gamma$. For small density departures, we show that with large $n$ they can be detected by the kernel test when $\\Delta_2$ is greater than $n^{- { 2 \\beta/( d + 4 \\beta ) }}$ up to a certain constant and $\\gamma$ scales as $n^{-1/(d+4\\beta)}$. The analysis extends to cases where the manifold has a boundary and the data samples contain high-dimensional additive noise. Our results indicate that the kernel two-sample test has no curse-of-dimensionality when the data lie on or near a low-dimensional manifold. We validate our theory and the properties of the kernel test for manifold data through a series of numerical experiments.","url_abs":"https://arxiv.org/abs/2105.03425v4","url_pdf":"https://arxiv.org/pdf/2105.03425v4.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":"kernel-mmd-two-sample-tests-for-manifold-data","repo_url":"https://github.com/xycheng/manifold_mmd","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"two","task_name":"Vocal Bursts Valence Prediction"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2105.03425","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}