Convergence of Laplacian Eigenmaps and Its Rate for Submanifolds with Singularities

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.