In this paper we improve the spectral convergence rates for graph-based approximations of weighted Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of the graph connectivity ε, our results show that the eigenvalues and eigenvectors of the graph Laplacian converge to those of a weighted Laplace-Beltrami operator at a rate of $O(n^{-1/(m+4)})$, up to log factors, where m is the manifold dimension and n is the number of vertices in the graph. Our approach is general and allows us to analyze a large variety of graph constructions that include ε-graphs and k-NN graphs. We also present the results of numerical experiments analyzing convergence rates on the two dimensional sphere.

在本文中，我们提高了从随机数据构建的加权 Laplace-Beltrami 算子的基于图的近似的谱收敛速度。我们利用连续特征函数的规律性和强的逐点一致性结果来证明谱收敛率与图拉普拉斯算子的逐点一致性率相同。特别是，对于图连通性ε的最佳选择，我们的结果表明，图拉普拉斯算子的特征值和特征向量以$○(n^{-1/(米+4)})$，直到对数因子，其中m是流形维度，n是图中的顶点数。我们的方法是通用的，允许我们分析包括ε -graph 和k -NN graphs 在内的大量图结构。我们还展示了分析二维球体收敛率的数值实验结果。