Corpus ID: 231698724

Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation

@article{Cheng2021EigenconvergenceOG,
  title={Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation},
  author={Xiuyuan Cheng and Nan Wu},
  journal={ArXiv},
  year={2021},
  volume={abs/2101.09875}
}
This work studies the spectral convergence of graph Laplacian to the Laplace-Beltrami operator when the graph affinity matrix is constructed from N random samples on a d-dimensional manifold embedded in a possibly high dimensional space. By analyzing Dirichlet form convergence and constructing candidate approximate eigenfunctions via convolution with manifold heat kernel, we prove that, with Gaussian kernel, one can set the kernel bandwidth parameter ∼ (logN/N) such that the eigenvalue… Expand

Figures and Tables from this paper

References

SHOWING 1-10 OF 37 REFERENCES
Spectral Convergence of the connection Laplacian from random samples
Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linearExpand
Convergence of Laplacian spectra from random samples
TLDR
It is proved that the eigenvectors and eigenvalues obtained by PIM converge in the limit of infinitely many random samples independently from a distribution (not necessarily to be uniform distribution). Expand
From graph to manifold Laplacian: The convergence rate
Abstract The convergence of the discrete graph Laplacian to the continuous manifold Laplacian in the limit of sample size N → ∞ while the kernel bandwidth e → 0 , is the justification for the successExpand
Lipschitz regularity of graph Laplacians on random data clouds
TLDR
This paper proves high probability interior and global Lipschitz estimates for solutions of graph Poisson equations, and obtains high probability and approximate convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators. Expand
Spectral Convergence Rate of Graph Laplacian
Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from aExpand
Uniform Convergence of Adaptive Graph-Based Regularization
TLDR
This paper identifies the limit of the regularizer and shows uniform convergence over the space of Holder functions, which are of independent interest for the theoretical analysis of manifold-based learning methods. Expand
Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator
TLDR
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. Expand
Manifold learning with bi-stochastic kernels
In this paper we answer the following question: what is the infinitesimal generator of the diffusion process defined by a kernel that is normalized such that it is bi-stochastic with respect to aExpand
From Graphs to Manifolds - Weak and Strong Pointwise Consistency of Graph Laplacians
TLDR
This paper establishes the strong pointwise consistency of a family of graph Laplacians with data-dependent weights to some weighted Laplace operator. Expand
Variable Bandwidth Diffusion Kernels
Practical applications of kernel methods often use variable bandwidth kernels, also known as self-tuning kernels, however much of the current theory of kernel based techniques is only applicable toExpand
...
1
2
3
4
...