Curvature-driven manifold fitting under unbounded isotropic noise

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 $\ddot{I}„$ from observations $Y=X+\hat{I}¾$, where $X$ is uniformly distributed on $\mathcal{M}$ and $\hat{I}¾\sim \mathcal{N}(0,\ddot{I}{ƒ}^2{I}_D)$ denotes isotropic Gaussian noise. To project any points $z$ in a tubular neighborhood $\hat{I}“$ of $\mathcal{M}$ onto $\mathcal{M}$, we construct a sample-based estimator $F:\hat{I}“\to {\mathbb{R}}^D$ by a normalized local kernel with the theoretically derived bandwidth $r=c_D\ddot{I}ƒ$. Under a sample size of $O(\ddot{I}{ƒ}^{-3d-5})$, 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 $\ddot{I}€(z)$ is the projection of $z$ onto $\mathcal{M}$ and $H_{\ddot{I}€(z)}$ is the mean curvature vector of $\mathcal{M}$ at $\ddot{I}€(z)$. The resulting manifold $F(\hat{I}“)$ has reach bounded below by $c\ddot{I}„$ for $c>0$ and achieves a state-of-the-art Hausdorff distance of $O(\ddot{I}{ƒ}^2)$ 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.

40 pages, 9 figures