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A Higher Order Local Mesh Method for Approximating 1-Laplacians on Unknown Manifolds

John Wilson Peoples, John Harlim

TL;DR

This work addresses the problem of approximating differential operators on vector fields defined on unknown manifolds embedded in high-dimensional space from random point samples. It introduces a local curved mesh method that combines local tangent-space projections with generalized moving least squares to produce curved local patches, enabling curvature-aware weak formulations for Bochner and Hodge Laplacians. The authors prove spectral convergence for the Bochner Laplacian and demonstrate strong numerical convergence on the sphere and torus, with robustness to moderate noise and out-of-sample evaluation. This approach avoids global meshing in high dimensions, provides a mesh-free, high-order, and scalable framework for vector-field operators on manifolds with practical importance for geometric data analysis.

Abstract

We introduce a numerical method for approximating arbitrary differential operators on vector fields in the weak form given point cloud data sampled randomly from a $d$ dimensional manifold embedded in $\mathbb{R}^n$. This method generalizes the local linear mesh method to the local curved mesh method, thus, allowing for the estimation of differential operators with nontrivial Christoffel symbols, such as the Bochner or Hodge Laplacians. In particular, we leverage the potentially small intrinsic dimension of the manifold $(d \ll n)$ to construct local parameterizations that incorporate both local meshes and higher-order curvature information. The former is constructed using low dimensional meshes obtained from local data projected to the tangent spaces, while the latter is obtained by fitting local polynomials with the generalized moving least squares. Theoretically, we prove the spectral convergence for the proposed method for the estimation of the Bochner Laplacian. We provide numerical results supporting the theoretical convergence rates for the Bochner and Hodge Laplacians on simple manifolds.

A Higher Order Local Mesh Method for Approximating 1-Laplacians on Unknown Manifolds

TL;DR

This work addresses the problem of approximating differential operators on vector fields defined on unknown manifolds embedded in high-dimensional space from random point samples. It introduces a local curved mesh method that combines local tangent-space projections with generalized moving least squares to produce curved local patches, enabling curvature-aware weak formulations for Bochner and Hodge Laplacians. The authors prove spectral convergence for the Bochner Laplacian and demonstrate strong numerical convergence on the sphere and torus, with robustness to moderate noise and out-of-sample evaluation. This approach avoids global meshing in high dimensions, provides a mesh-free, high-order, and scalable framework for vector-field operators on manifolds with practical importance for geometric data analysis.

Abstract

We introduce a numerical method for approximating arbitrary differential operators on vector fields in the weak form given point cloud data sampled randomly from a dimensional manifold embedded in . This method generalizes the local linear mesh method to the local curved mesh method, thus, allowing for the estimation of differential operators with nontrivial Christoffel symbols, such as the Bochner or Hodge Laplacians. In particular, we leverage the potentially small intrinsic dimension of the manifold to construct local parameterizations that incorporate both local meshes and higher-order curvature information. The former is constructed using low dimensional meshes obtained from local data projected to the tangent spaces, while the latter is obtained by fitting local polynomials with the generalized moving least squares. Theoretically, we prove the spectral convergence for the proposed method for the estimation of the Bochner Laplacian. We provide numerical results supporting the theoretical convergence rates for the Bochner and Hodge Laplacians on simple manifolds.
Paper Structure (24 sections, 10 theorems, 164 equations, 5 figures)

This paper contains 24 sections, 10 theorems, 164 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Local Metric Learning
  3. Local tangent space approximation
  4. Approximation of metric tensor with linear local meshes
  5. Generalized moving least squares approximation
  6. Approximation of metric tensor with local curved meshes
  7. Operator Approximation in Weak Form
  8. Weak approximation of the Laplace-Beltrami operator
  9. Generalization to operators on arbitrary tensor fields
  10. Convergence of Eigenvalues
  11. Local metric approximation
  12. Variational approximation
  13. Local vector field approximation
  14. Main result
  15. Numerical Results
  16. ...and 9 more sections

Key Result

Lemma 4.1

Let $X \subseteq M$ be such that $h_{X, M} \leq h_0$ for some constant $h_0 > 0$. Define $\Omega_i^* = \bigcup_{\vec{v} \in B_{\vec{0}}(r_i)} B_{\vec{v}}(C_2 h_0)$ for some constant $C_2 > 0$. Then for any $f \in C^{m+1}(\Omega_i^*)$ and a multi index $\ell$ that satisfies $|\ell| \leq m$, we have for any $\vec{v} \in B_{\vec{0}}(r_i)$, where $\hat{f}$ is a GMLS approximation to $f$ using polynom

Figures (5)

  • Figure 1: Example datasets used for numerical experiments. Displayed is a single trial with $N=4000$ points for (a) sphere (b) torus, and (c) noisy sphere (each data point is perturbed in the normal direction with uniform noise of size $0.10$).
  • Figure 2: Operator estimation on sphere with uniform sampling distribution. Average of relative error (eigenvalues, leading $L=48$ modes) and Average of mean-square-error (eigenvectors, leading $L=6$ modes) as defined in Equation \ref{['eigenvalue-metric']}, \ref{['eigenvector-metric']} respectively: (a) Bochner Laplacian; (b) Hodge Laplacian.
  • Figure 3: Visual comparison of the first four eigenvector fields for the Bochner Laplacian obtained using CMM (row 1) and analytic methods (row 2). The estimates are obtained from a single trial of $N=6000$ randomly sampled points. Color indicates vector norm. Vectors visualized correspond to on a random subset of size $2000$.
  • Figure 4: Operator estimation on torus with uniform sampling (with rejection) distribution. (a) Average of relative error (over the first $10$ nontrivial) eigenvalues of the Hodge Laplacian estimator obtained from CMM as a function of $N$. (b) Mode by mode estimation for of spectrum for a single trial with $N=16000$.
  • Figure 5: Operator estimation on a noisy sphere with uniform sampling distribution as functions of $N$. (a) Mean of percent error of eigenvalues. (b) Mean of mean-square-error of eigenvectors. Metrics are as defined in Equations \ref{['eigenvalue-metric']} and \ref{['eigenvector-metric']} Color indicates noise level.

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 4.1
  • Lemma 4.2
  • Definition 4.1
  • Lemma 4.3
  • proof
  • Definition 4.2: Ritz projection
  • Lemma 4.4
  • ...and 11 more