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 $\epsilon \sim (\log N/ N)^{1/(d/2+2)}$ such that the eigenvalue convergence rate is $N^{-1/(d/2+2)}$ and the eigenvector convergence in 2-norm has rate $N^{-1/(d+4)}$; When $\epsilon \sim N^{-1/(d/2+3)}$, both eigenvalue and eigenvector rates are $N^{-1/(d/2+3)}$. These rates are up to a $\log N$ factor and proved for finitely many low-lying eigenvalues. The result holds for un-normalized and random-walk graph Laplacians when data are uniformly sampled on the manifold, as well as the density-corrected graph Laplacian (where the affinity matrix is normalized by the degree matrix from both sides) with non-uniformly sampled data. As an intermediate result, we prove new point-wise and Dirichlet form convergence rates for the density-corrected graph Laplacian. Numerical results are provided to verify the theory.

翻译：这项工作研究图形 Laplacecian 与 Laplace- Beltrami 操作器的光谱融合, 当图形亲近性矩阵由位于可能高维空间内嵌的以美元为单位的以美元为单位的随机样本构建时 。 通过分析 Dirichlet 的趋同形式, 并通过与多个热内核相融合来构建候选的近似电子元元功能, 我们证明, 有了高斯 内核, 就可以设置内核带带参数$\ epsilon\sim (\ log N/ N) 1/ (d/2+2)} 。 因此, 以美元为单位的中值递增值递增率是 $_ 1 / (d/2+2) = = $, 以美元为单位的中值为单位, 以正值为单位的内值为单位的内值为内值, 以正值为内值的内基值为内基数的内基值, 以内基值为内基数的内基数的内基值为内基值数据的内基值。