Curvature-driven manifold fitting under unbounded isotropic noise

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 $\mathrm{\Gamma }$ of $\mathcal{M}$ onto $\mathcal{M}$, we construct a sample-based estimator $F:\mathrm{\Gamma }\to {\mathbb{R}}^D$ 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 $$F(z)=\pi (z)+\frac{d}{2}{H}_{\pi (z)}{\sigma }^2+O({\sigma }^3),z\in \mathrm{\Gamma },$$ 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 $F(\mathrm{\Gamma })$ 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.