Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities

• URL: http://arxiv.org/abs/2110.08138v1
• Date: Fri, 15 Oct 2021 15:06:44 GMT
• Title: Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities
• Authors: Masayuki Aino
• Abstract summary: 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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