Curvature-driven manifold fitting under unbounded isotropic noise

TL;DR

This work addresses recovering a low-dimensional manifold $\\mathcal{M}\\subset\\mathbb{R}^D$ from noisy samples $Y=X+\\xi$ with Gaussian ambient noise. It introduces a curvature-driven local mean-shift estimator $F$ with bandwidth $r=c_D\\sigma$ that directly inverts the noise-generating process, yielding a population expansion $F(z)=\\pi(z)+\frac{d}{2}H_{\\pi(z)}\\sigma^2+O(\\sigma^3)$ and a uniformly close output manifold with Hausdorff error $O(\\sigma^2)$ and a guaranteed reach proportional to the true reach $\\tau$. The main theoretical contributions include a detailed density and score expansion of $p_{\\sigma}$ near the manifold, a precise link between the normalized graph operator and mean curvature flow, and high-probability sample-size bounds $N=O(\\sigma^{-3d-5})$ for uniform consistency. Numerically, the method demonstrates quadratic error decay and superior efficiency across circles, tori, and Calabi–Yau-like manifolds, supporting its practical viability for curvature-aware manifold denoising and reconstruction.

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 $τ$ 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.

Figures (9)

• Figure 1.1: The mean curvature $H$ of a saddle.
• Figure 1.2: The figure illustrates samples corrupted by isotropic noise. Points $x \in \mathcal{M}$ on the underlying submanifold are displaced by isotropic noise $\sigma$ (blue dashed circles), and the dominant bias of the density of noise appears in the normal direction (red dashed lines). At $x$, $T_x \mathcal{M}$, $T_x^\bot \mathcal{M}$ and $H_x$ denote the tangent space, normal space and mean curvature vector respectively. At scale $r$, $\mathcal{M}$ deviates from $T_x \mathcal{M}$ in the normal direction by order $O(r^2)$.
• Figure 1.3: (a) Local PCA: For an observed point $y$, its projection $\pi(y)$ on $\mathcal{M}$, the tangent space $T_{\pi(y)}\mathcal{M}$, and the estimated tangent space $\widehat{{T}_{y}\mathcal{M}}$ constructed from neighboring points $y_i, y_j$; (b) Local averaging: The average points $ave_y$ of observed samples (black dots) within a spherical (blue) or cylindrical (red) neighborhood centered at $y$.
• Figure 1.4: An illustration for constructing the estimator. The underlying submanifold $\mathcal{M}$, the estimator $F$ at point $z$ is constructed by the weighted average of noisy samples (black dots) lying within a distance $r$ from $z$. Let $\pi(z)$ denote the projection of $z$ onto $\mathcal{M}$, and define the vector $v_z = z - \pi(z)$.
• Figure 2.1: Exponential map and its Taylor expansions.

Theorems & Definitions (48)

• Theorem 1.1
• Remark 1.2
• Definition 2.1: Reach
• Lemma 2.2: Federer's reach condition
• Lemma 2.3: Niyogi2008FindingTH, Proposition 6.1
• Lemma 2.4: Alexander2005GaussEA Corollary 1.4
• Lemma 2.5
• Lemma 2.6
• Definition 2.7: Hausdorff distance
• Lemma 2.8