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 $\epsilon \sim (\log N/N)^{1/(d/2+3)}$, 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 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中嵌入的美元方元元元元体的随机样本构建时, 图形亲近性矩阵的光谱融合。 通过分析 Dirichlet 的趋同形式, 并通过多热内核熔化来构建候选的近似电子元元元元元元元元元元功能; 当 Gaussian 内核( log N/ N) 1/ (d/2+3) 时, 我们证明, 有了 Gausian 内核素带带带的内核带带带带参数 $\ epsilon \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( d \ \ \ \ \ \ \ \ \ \ \ \ \ ( d \ \ \ \ \ \ \ \ \ \ \ \ \ \ ) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \