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.
Categories: math.ST stat.TH