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

@inproceedings{Aino2021ConvergenceOL, title={Convergence of Laplacian Eigenmaps and its Rate for Submanifolds with Singularities}, author={Masayuki Aino}, year={2021} }

In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the ǫneighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is O (

## References

SHOWING 1-10 OF 26 REFERENCES

A graph discretization of the Laplace-Beltrami operator

- Mathematics
- 2013

We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator…

Convergence of Laplacian Eigenmaps

- Computer Science, MathematicsNIPS
- 2006

This paper shows convergence of eigenvectors of the point cloud Laplacian to the eigenfunctions of the Laplace-Beltrami operator on the underlying manifold, thus establishing the first convergence results for a spectral dimensionality reduction algorithm in the manifold setting.

Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering

- Computer Science, MathematicsNIPS
- 2001

The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering.

Embeddings of Riemannian manifolds with heat kernels and eigenfunctions

- Mathematics
- 2013

We show that any closed n-dimensional Riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points. Both this time and this number can…

Degeneration of Riemannian metrics under Ricci curvature bounds

- Mathematics
- 2013

These notes are based on the Fermi Lectures delivered at the Scuola Normale Superiore, Pisa, in June 2001. The principal aim of the lectures was to present the structure theory developed by Toby…

The embedding dimension of Laplacian eigenfunction maps

- Mathematics, Computer ScienceArXiv
- 2016

The maximal embedding dimension of any closed, connected Riemannian manifold can be smoothly embedded by its Laplacian eigenfunction maps into $\mathbb{R}^m$ for some $m$.

Finding the Homology of Submanifolds with High Confidence from Random Samples

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2008

This work considers the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space and shows how to “learn” the homology of the sub manifold with high confidence.

On the structure of spaces with Ricci curvature bounded below. II

- Mathematics
- 2000

In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de…

Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator

- Mathematics, Computer ScienceFound. Comput. Math.
- 2020

The convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m -dimensional submanifold M in R d is studied as the sample size n increases and the neighborhood size h tends to zero.

Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions

- Mathematics
- 2018

In this paper we study the family of embeddings $\Phi_t$ of a compact $RCD^*(K,N)$ space $(X,d,m)$ into $L^2(X,m)$ via eigenmaps. Extending part of the classical results by Berard,…