基于流形热插值的 Gaussian 核化图 Laplacian 的特征值收敛性
Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation
AI总结：
本研究通过分析 Dirichlet 形式收敛性并利用流形热核卷积构建近似特征函数，证明了 Gaussian 核化图 Laplacian 向 Laplace-Beltrami 算子的特征值与特征向量收敛率，适用于多种采样与归一化设置。
摘要：
本研究探讨了当图亲和矩阵由嵌入在可能高维空间中的 $d$ 维流形上的 $N$ 个随机样本构建时，图 Laplacian 向 Laplace-Beltrami 算子的谱收敛性。通过分析 Dirichlet 形式收敛性，并利用与流形热核的卷积构建候选近似特征函数，我们证明了：在使用 Gaussian 核时，可设置核带宽参数 $ε\sim (\log N/ N)^{1/(d/2+2)}$，使得特征值收敛率为 $N^{-1/(d/2+2)}$，且特征向量在 2-范数下的收敛率为 $N^{-1/(d+4)}$；当 $ε\sim (\log N/N)^{1/(d/2+3)}$ 时，特征值与特征向量的收敛率均为 $N^{-1/(d/2+3)}$。这些收敛率受 $\log N$ 因子影响，且是针对有限个低阶特征值证明的。该结果适用于数据在流形上均匀采样时的非归一化和随机游走图 Laplacian，以及采用非均匀采样数据的密度校正图 Laplacian（其亲和矩阵由度矩阵从两侧归一化）。作为中间结果，我们证明了密度校正图 Laplacian 的新逐点和 Dirichlet 形式收敛率。提供了数值结果以验证该理论。
英文摘要：
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 $ε\sim (\log N/ N)^{1/(d/2+2)}$ 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 $ε\sim (\log N/N)^{1/(d/2+3)}$, 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.
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