{"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/density-estimation-on-an-unknown-submanifold","title":"Density estimation on an unknown submanifold","arxiv_id":"1910.08477","date":"2019-10-18","proceeding":null,"authors":["Clément Berenfeld","Marc Hoffmann"],"abstract":"We investigate density estimation from a $n$-sample in the Euclidean space $\\mathbb R^D$, when the data is supported by an unknown submanifold $M$ of possibly unknown dimension $d < D$ under a reach condition. We study nonparametric kernel methods for pointwise loss, with data-driven bandwidths that incorporate some learning of the geometry via a local dimension estimator. When $f$ has H\\\"older smoothness $\\beta$ and $M$ has regularity $\\alpha$, our estimator achieves the rate $n^{-\\alpha \\wedge \\beta/(2\\alpha \\wedge \\beta+d)}$ and does not depend on the ambient dimension $D$ and is asymptotically minimax for $\\alpha \\geq \\beta$. Following Lepski's principle, a bandwidth selection rule is shown to achieve smoothness adaptation. We also investigate the case $\\alpha \\leq \\beta$: by estimating in some sense the underlying geometry of $M$, we establish in dimension $d=1$ that the minimax rate is $n^{-\\beta/(2\\beta+1)}$ proving in particular that it does not depend on the regularity of $M$. Finally, a numerical implementation is conducted on some case studies in order to confirm the practical feasibility of our estimators.","url_abs":"http://arxiv.org/abs/1910.08477v2","url_pdf":"http://arxiv.org/pdf/1910.08477v2.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":"density-estimation-on-an-unknown-submanifold","repo_url":"https://github.com/sverdoot/density-estimation-unknown-submanifold","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","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}