Curvature-driven manifold fitting under unbounded isotropic noise
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866909991023149056 |
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| author | Li, Ruowei Yao, Zhigang |
| author_facet | Li, Ruowei Yao, Zhigang |
| contents | 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 $τ$ from observations $Y = X + ξ$, where $X$ is uniformly distributed on $\mathcal{M}$ and $ξ\sim \mathcal{N}(0, σ^2 I_D)$ denotes isotropic Gaussian noise. To project any points $z$ in a tubular neighborhood $Γ$ of $\mathcal{M}$ onto $\mathcal{M}$, we construct a sample-based estimator $F:Γ\to\mathbb{R}^D$ by a normalized local kernel with the theoretically derived bandwidth $r = c_Dσ$. Under a sample size of $O(σ^{-3d-5})$, we establish with high probability the uniform asymptotic expansion \[ F(z) = π(z) + \frac{d}{2} H_{π(z)} σ^2 + O(σ^3), \qquad z \in Γ, \] where $π(z)$ is the projection of $z$ onto $\mathcal{M}$ and $H_{π(z)}$ is the mean curvature vector of $\mathcal{M}$ at $π(z)$. The resulting manifold $F(Γ)$ has reach bounded below by $c τ$ for $c>0$ and achieves a state-of-the-art Hausdorff distance of $O(σ^2)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10133 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Curvature-driven manifold fitting under unbounded isotropic noise Li, Ruowei Yao, Zhigang Statistics Theory 62R30, 53Z50 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 $τ$ from observations $Y = X + ξ$, where $X$ is uniformly distributed on $\mathcal{M}$ and $ξ\sim \mathcal{N}(0, σ^2 I_D)$ denotes isotropic Gaussian noise. To project any points $z$ in a tubular neighborhood $Γ$ of $\mathcal{M}$ onto $\mathcal{M}$, we construct a sample-based estimator $F:Γ\to\mathbb{R}^D$ by a normalized local kernel with the theoretically derived bandwidth $r = c_Dσ$. Under a sample size of $O(σ^{-3d-5})$, we establish with high probability the uniform asymptotic expansion \[ F(z) = π(z) + \frac{d}{2} H_{π(z)} σ^2 + O(σ^3), \qquad z \in Γ, \] where $π(z)$ is the projection of $z$ onto $\mathcal{M}$ and $H_{π(z)}$ is the mean curvature vector of $\mathcal{M}$ at $π(z)$. The resulting manifold $F(Γ)$ has reach bounded below by $c τ$ for $c>0$ and achieves a state-of-the-art Hausdorff distance of $O(σ^2)$ 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. |
| title | Curvature-driven manifold fitting under unbounded isotropic noise |
| topic | Statistics Theory 62R30, 53Z50 |
| url | https://arxiv.org/abs/2601.10133 |