# 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 ϵ-neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is O((log n/n)^1/(m+2)), where m and n denote the dimension of the manifold and the sample size, respectively.

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