dorsal/arxiv
View SchemaCurvature-driven manifold fitting under unbounded isotropic noise
| Authors | Ruowei Li, Zhigang Yao |
|---|---|
| Categories | |
| ArXiv ID | 2601.10133vv1 |
| URL | https://arxiv.org/abs/2601.10133 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
Manifold fitting aims to reconstruct a low-dimensional manifold from high-dimensional data, whose framework is established by Fefferman et al. \cite{fefferman2020reconstruction,fefferman2021reconstruction}. This paper studies the recovery of a compact $C^3$ submanifold $\mathcal{M} \subset \mathbb{R}^D$ with dimension $d<D$ and positive reach $\tau$ from observations $Y = X + \xi$, where $X$ is uniformly distributed on $\mathcal{M}$ and $\xi \sim \mathcal{N}(0, \sigma^2 I_D)$ denotes isotropic Gaussian noise. To project any points $z$ in a tubular neighborhood $\Gamma$ of $\mathcal{M}$ onto $\mathcal{M}$, we construct a sample-based estimator $F:\Gamma\to\mathbb{R}^D$ by a normalized local kernel with the theoretically derived bandwidth $r = c_D\sigma$. Under a sample size of $O(\sigma^{-3d-5})$, we establish with high probability the uniform asymptotic expansion \[ F(z) = \pi(z) + \frac{d}{2} H_{\pi(z)} \sigma^2 + O(\sigma^3), \qquad z \in \Gamma, \] where $\pi(z)$ is the projection of $z$ onto $\mathcal{M}$ and $H_{\pi(z)}$ is the mean curvature vector of $\mathcal{M}$ at $\pi(z)$. The resulting manifold $F(\Gamma)$ has reach bounded below by $c \tau$ for $c>0$ and achieves a state-of-the-art Hausdorff distance of $O(\sigma^2)$ to $\mathcal{M}$. Numerical experiments confirm the quadratic decay of the reconstruction error and demonstrate the computational efficiency of the estimator $F$. Our work provides a curvature-driven framework for denoising and reconstructing manifolds with second-order accuracy.
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"abstract": "Manifold fitting aims to reconstruct a low-dimensional manifold from high-dimensional data, whose framework is established by Fefferman et al. \\cite{fefferman2020reconstruction,fefferman2021reconstruction}. This paper studies the recovery of a compact $C^3$ submanifold $\\mathcal{M} \\subset \\mathbb{R}^D$ with dimension $d\u003cD$ and positive reach $\\tau$ from observations $Y = X + \\xi$, where $X$ is uniformly distributed on $\\mathcal{M}$ and $\\xi \\sim \\mathcal{N}(0, \\sigma^2 I_D)$ denotes isotropic Gaussian noise. To project any points $z$ in a tubular neighborhood $\\Gamma$ of $\\mathcal{M}$ onto $\\mathcal{M}$, we construct a sample-based estimator $F:\\Gamma\\to\\mathbb{R}^D$ by a normalized local kernel with the theoretically derived bandwidth $r = c_D\\sigma$. Under a sample size of $O(\\sigma^{-3d-5})$, we establish with high probability the uniform asymptotic expansion \\[ F(z) = \\pi(z) + \\frac{d}{2} H_{\\pi(z)} \\sigma^2 + O(\\sigma^3), \\qquad z \\in \\Gamma, \\] where $\\pi(z)$ is the projection of $z$ onto $\\mathcal{M}$ and $H_{\\pi(z)}$ is the mean curvature vector of $\\mathcal{M}$ at $\\pi(z)$. The resulting manifold $F(\\Gamma)$ has reach bounded below by $c \\tau$ for $c\u003e0$ and achieves a state-of-the-art Hausdorff distance of $O(\\sigma^2)$ to $\\mathcal{M}$. Numerical experiments confirm the quadratic decay of the reconstruction error and demonstrate the computational efficiency of the estimator $F$. Our work provides a curvature-driven framework for denoising and reconstructing manifolds with second-order accuracy.",
"arxiv_id": "2601.10133",
"authors": [
"Ruowei Li",
"Zhigang Yao"
],
"categories": [
"math.ST",
"stat.TH"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Curvature-driven manifold fitting under unbounded isotropic noise",
"url": "https://arxiv.org/abs/2601.10133",
"version": "v1"
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