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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.08138
 Bibcode:
 2021arXiv211008138A
 Keywords:

 Mathematics  Differential Geometry;
 Statistics  Machine Learning;
 58C40;
 58J50;
 60D05
 EPrint:
 63 pages