# Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities

arXiv: https://arxiv.org/abs/2110.08138 · HTML: https://mlsanity.com/paper/2110.08138/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 22 references that are in the map.
- First posted on arXiv 15 Oct 2021.

## Builds on (most cited first, 4 of 4)

- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs](https://mlsanity.com/paper/1910.13476/) (2019)
- [The embedding dimension of Laplacian eigenfunction maps](https://mlsanity.com/paper/1605.01643/) (2016)
- [Four lectures on probabilistic methods for data science](https://mlsanity.com/paper/1612.06661/) (2016)

## Abstract (arXiv)

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 Oleft(left(log n/nright)1/(m+2)right), where m and n denote the dimension of the manifold and the sample size, respectively.