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Spectral Convergence of Symmetrized Graph Laplacian on manifolds with boundary

J. Wilson Peoples, John Harlim

TL;DR

We address the problem of spectral convergence for a symmetrized Gaussian graph Laplacian constructed from data on a manifold with boundary. The authors develop a unified RKHS-based min-max framework to relate discrete eigenpairs to Laplace-Beltrami eigenpairs under both Neumann and Dirichlet boundary conditions, with Dirichlet convergence achieved via a Truncated Graph Laplacian on interior points. They provide explicit convergence rates for eigenvalues and eigenvectors, demonstrating Neumann rates of the form $\mathcal{O}\left( \frac{\sqrt{\log n}}{\epsilon^{d/2+1} \sqrt{n}} + \epsilon^{1/2} \right)$ and Dirichlet rates involving a truncation parameter $\gamma$, along with $L^2$-eigenvector convergence. Numerical experiments on a semi-circle and a semi-torus corroborate the theory and illustrate practical efficacy of truncation for Dirichlet problems, highlighting the method’s relevance for PDEs on unknown manifolds and diffusion-map-based representations.

Abstract

We study the spectral convergence of a symmetrized Graph Laplacian matrix induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from a manifold with boundary, a sub-manifold of $\mathbb{R}^m$. Specifically, we deduce the convergence rates for eigenpairs of the discrete Graph-Laplacian matrix to the eigensolutions of the Laplace-Beltrami operator that are well-defined on manifolds with boundary, including the homogeneous Neumann and Dirichlet boundary conditions. For the Dirichlet problem, we deduce the convergence of the \emph{truncated Graph Laplacian}, which is recently numerically observed in applications, and provide a detailed numerical investigation on simple manifolds. Our method of proof relies on the min-max argument over a compact and symmetric integral operator, leveraging the RKHS theory for spectral convergence of integral operator and a recent pointwise asymptotic result of a Gaussian kernel integral operator on manifolds with boundary.

Spectral Convergence of Symmetrized Graph Laplacian on manifolds with boundary

TL;DR

We address the problem of spectral convergence for a symmetrized Gaussian graph Laplacian constructed from data on a manifold with boundary. The authors develop a unified RKHS-based min-max framework to relate discrete eigenpairs to Laplace-Beltrami eigenpairs under both Neumann and Dirichlet boundary conditions, with Dirichlet convergence achieved via a Truncated Graph Laplacian on interior points. They provide explicit convergence rates for eigenvalues and eigenvectors, demonstrating Neumann rates of the form and Dirichlet rates involving a truncation parameter , along with -eigenvector convergence. Numerical experiments on a semi-circle and a semi-torus corroborate the theory and illustrate practical efficacy of truncation for Dirichlet problems, highlighting the method’s relevance for PDEs on unknown manifolds and diffusion-map-based representations.

Abstract

We study the spectral convergence of a symmetrized Graph Laplacian matrix induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from a manifold with boundary, a sub-manifold of . Specifically, we deduce the convergence rates for eigenpairs of the discrete Graph-Laplacian matrix to the eigensolutions of the Laplace-Beltrami operator that are well-defined on manifolds with boundary, including the homogeneous Neumann and Dirichlet boundary conditions. For the Dirichlet problem, we deduce the convergence of the \emph{truncated Graph Laplacian}, which is recently numerically observed in applications, and provide a detailed numerical investigation on simple manifolds. Our method of proof relies on the min-max argument over a compact and symmetric integral operator, leveraging the RKHS theory for spectral convergence of integral operator and a recent pointwise asymptotic result of a Gaussian kernel integral operator on manifolds with boundary.

Paper Structure

This paper contains 19 sections, 28 theorems, 233 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Discrete and Continuous Estimators
  3. Laplace-Beltrami and Kernel Integral Operators
  4. Discretized Integral Operators
  5. Spectral Convergence Results for Neumann Boundary Conditions
  6. Spectral Convergence Results for Dirichlet Boundary Conditions
  7. Numerical Results for Dirichlet Boundary Conditions
  8. Dirichlet Laplacian on a Semi-Circle
  9. Dirichlet Laplacian on a Semi-Torus
  10. Summary
  11. Interpolation and Restriction via RKHS Theory
  12. Lemma for Spectral Convergence
  13. Proof of Theorem \ref{['conveigvec']}
  14. Proof of Lemma \ref{['truncated matrix eigenvalues']}
  15. Convergence of Dirichlet Eigenvectors
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $\mathcal{M}$ be a compact, smooth manifold either without boundary, or with a $C^3$ boundary and normal collar. Let $\epsilon>0$, then for $f, \phi \in C^\infty(\mathcal{M})$ as $\epsilon\to 0$.

Figures (7)

  • Figure 1: Semi-Circle Example with $n=10000$: (a) Absolute errors of eigenvalues as functions of mode $k$; (b) Root-mean-square errors of eigenvectors as functions of mode $k$; (c) Estimated eigenvectors from well-sampled data, normalized with $m_o^\partial$; (d) Estimated eigenvectors from random data, normalized with $m_o^\partial$. In panels (a) and (b), we show estimates from 10 random realizations (black curves).
  • Figure 2: Semi-Circle Example for estimation with $m_o^\partial$ normalization, uniform sampling distribution. Mean of relative error (eigenvalues) and mean of mean-square-error (eigenvectors) as defined in Equation \ref{['empiricalerror']} over the first $M=10$ modes as functions of $n$: (a) Well-sampled data; (b) Random data.
  • Figure 3: Semi-Circle Example for estimation with $m_o^\partial$ normalization, nonuniform sampling distribution. Mean of relative error (eigenvalues) and mean of mean-square-error (eigenvectors) as defined in Equation \ref{['empiricalerror']} over the first $M=10$ modes as functions of $n$: (a) Well-sampled data; (b) Random data.
  • Figure 4: Semicircle example: Estimated leading eigenvalue as a function of $\gamma$. There are 10 trials for each $\gamma$, the estimate $\tilde{\lambda}_1^{\gamma,\epsilon,n}$ for each trial is denoted in red 'x'. The blue curve denotes the average of these estimates.
  • Figure 5: Semi-Torus Example for estimation with $m^\partial_0$ normalization, uniform sampling distribution. Mean of relative error (eigenvalues) and mean of mean-square-error (eigenvectors) as defined in Equation \ref{['empiricalerror']} as functions of $n$: (a) Over the first $M=10$ modes, well-sampled data; (b) Over the first $M=3$ modes, random data.
  • ...and 2 more figures

Theorems & Definitions (59)

  • Definition 2.1
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['weak convergence']}
  • Theorem 3.1
  • Remark 3.1
  • proof
  • Theorem 3.2
  • Theorem 4.1
  • Definition 4.1
  • Lemma 4.1
  • ...and 49 more