{"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/non-parametric-estimation-of-manifolds-from","title":"Non-Parametric Estimation of Manifolds from Noisy Data","arxiv_id":"2105.04754","date":"2021-05-11","proceeding":null,"authors":["Yariv Aizenbud","Barak Sober"],"abstract":"A common observation in data-driven applications is that high dimensional data has a low intrinsic dimension, at least locally. In this work, we consider the problem of estimating a $d$ dimensional sub-manifold of $\\mathbb{R}^D$ from a finite set of noisy samples. Assuming that the data was sampled uniformly from a tubular neighborhood of $\\mathcal{M}\\in \\mathcal{C}^k$, a compact manifold without boundary, we present an algorithm that takes a point $r$ from the tubular neighborhood and outputs $\\hat p_n\\in \\mathbb{R}^D$, and $\\widehat{T_{\\hat p_n}\\mathcal{M}}$ an element in the Grassmanian $Gr(d, D)$. We prove that as the number of samples $n\\to\\infty$ the point $\\hat p_n$ converges to $p\\in \\mathcal{M}$ and $\\widehat{T_{\\hat p_n}\\mathcal{M}}$ converges to $T_p\\mathcal{M}$ (the tangent space at that point) with high probability. Furthermore, we show that the estimation yields asymptotic rates of convergence of $n^{-\\frac{k}{2k + d}}$ for the point estimation and $n^{-\\frac{k-1}{2k + d}}$ for the estimation of the tangent space. These rates are known to be optimal for the case of function estimation.","url_abs":"https://arxiv.org/abs/2105.04754v2","url_pdf":"https://arxiv.org/pdf/2105.04754v2.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":"non-parametric-estimation-of-manifolds-from","repo_url":"https://github.com/aizeny/manapprox","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"2k","task_name":"2k"}],"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}