Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities

2021/10/15 by Aino, Masayuki
#58C40 #58J50 #60D05 #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML)

paper · doi:10.48550/arxiv.2110.08138

Abstract

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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