Convergence of Laplacian Eigenmaps and Its Rate for Submanifolds with Singularities

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