Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation

TL;DR

This paper establishes spectral convergence rates of Gaussian kernel-based graph Laplacians to the Laplace-Beltrami operator on manifolds, providing theoretical guarantees for eigenvalues and eigenvectors with different kernel bandwidths.

Contribution

It introduces new convergence rates for eigenvalues and eigenvectors of graph Laplacians constructed from random samples on manifolds, including density-corrected variants, with rigorous proofs.

Findings

Eigenvalue convergence rate is approximately N^{-1/(d/2+2)} for a specific kernel bandwidth.

Eigenvector convergence in 2-norm is approximately N^{-1/(d/2+4)} under certain conditions.

Theoretical results are validated with numerical experiments.

Abstract

This work studies the spectral convergence of graph Laplacian to the Laplace-Beltrami operator when the graph affinity matrix is constructed from $N$ random samples on a $d$-dimensional manifold embedded in a possibly high dimensional space. By analyzing Dirichlet form convergence and constructing candidate approximate eigenfunctions via convolution with manifold heat kernel, we prove that, with Gaussian kernel, one can set the kernel bandwidth parameter $\epsilon \sim (\log N/ N)^{1/(d/2+2)}$ such that the eigenvalue convergence rate is $N^{-1/(d/2+2)}$ and the eigenvector convergence in 2-norm has rate $N^{-1/(d+4)}$; When $\epsilon \sim (\log N/N)^{1/(d/2+3)}$, both eigenvalue and eigenvector rates are $N^{-1/(d/2+3)}$. These rates are up to a $\log N$ factor and proved for finitely many low-lying eigenvalues. The result holds for un-normalized and random-walk graph Laplacians when data are uniformly sampled on the manifold, as well as the density-corrected graph Laplacian (where the affinity matrix is normalized by the degree matrix from both sides) with non-uniformly sampled data. As an intermediate result, we prove new point-wise and Dirichlet form convergence rates for the density-corrected graph Laplacian. Numerical results are provided to verify the theory.