We study the spectral convergence of graph Laplacians to the Laplace-Beltrami operator when the kernelized graph affinity matrix is constructed from N random samples on a d-dimensional manifold in an ambient Euclidean space. By analyzing Dirichlet form convergence and constructing candidate approximate eigenfunctions via convolution with manifold heat kernel, we prove eigen-convergence with rates as N increases. The best eigenvalue convergence rate is $N^{-1/(d/2+2)}$ (when the kernel bandwidth parameter $ϵ\sim {(\log N/N)}^{1/(d/2+2)}$) and the best eigenvector 2-norm convergence rate is $N^{-1/(d/2+3)}$ (when $ϵ\sim {(\log N/N)}^{1/(d/2+3)}$). These rates hold up to a $\log N$-factor for finitely many low-lying eigenvalues of both un-normalized and normalized graph Laplacians. When data density is non-uniform, we prove the same rates for the density-corrected graph Laplacian, and we also establish new operator point-wise convergence rate and Dirichlet form convergence rate as intermediate results. Numerical results are provided to support the theory.

当核化图亲和矩阵由环境欧几里得空间中的d维流形上的N个随机样本构造时，我们研究了图拉普拉斯算子与拉普拉斯-贝尔特拉米算子的谱收敛性。通过分析狄利克雷形式的收敛性并通过与流形热核的卷积构造候选近似特征函数，我们证明了特征收敛性随着N增加的速率。最佳特征值收敛速度为${ñ}^{-1/(d/2+2)}$（当内核带宽参数$\epsilon ～{(\mathrm{日志}ñ/ñ)}^{1/(d/2+2)}$) 和最佳特征向量 2-范数收敛速度是${ñ}^{-1/(d/2+3)}$（什么时候$\epsilon ～{(\mathrm{日志}ñ/ñ)}^{1/(d/2+3)}$）。这些费率高达$\mathrm{日志}ñ$- 非归一化和归一化图拉普拉斯算子的有限多个低位特征值的因子。当数据密度不均匀时，我们证明了密度校正图拉普拉斯算子的相同速率，并且我们还建立了新的算子逐点收敛速度和狄利克雷形式收敛速度作为中间结果。提供了数值结果来支持该理论。