Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities

TL;DR

This paper establishes a spectral approximation result for the Laplacian on singular submanifolds using random point samples, providing a convergence rate for eigenvalues based on sample size and manifold dimension.

Contribution

It introduces a convergence rate for Laplacian eigenvalues on singular submanifolds using neighborhood graphs from random samples, extending spectral approximation theory.

Findings

Convergence rate of eigenvalues is $O((rac{ extlog n}{n})^{1/(m+2)})$

Spectral approximation holds for submanifolds with singularities

Provides theoretical foundation for data-driven spectral analysis on complex manifolds

Abstract

In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the $\epsilon$-neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is $O\left(\left(\log n/n\right)^{1/(m+2)}\right)$, where $m$ and $n$ denote the dimension of the manifold and the sample size, respectively.