Emergent Mind
Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities
(2110.08138)
Published Oct 15, 2021
in
math.DG
and
stat.ML
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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