We study the spectral convergence of a symmetrized Graph Laplacian matrix induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from a manifold with boundary, a sub-manifold of $\mathbb{R}^m$. Specifically, we deduce the convergence rates for eigenpairs of the discrete Graph-Laplacian matrix to the eigensolutions of the Laplace-Beltrami operator that are well-defined on manifolds with boundary, including the homogeneous Neumann and Dirichlet boundary conditions. For the Dirichlet problem, we deduce the convergence of the \emph{truncated Graph Laplacian}, which is recently numerically observed in applications, and provide a detailed numerical investigation on simple manifolds. Our method of proof relies on the min-max argument over a compact and symmetric integral operator, leveraging the RKHS theory for spectral convergence of integral operator and a recent pointwise asymptotic result of a Gaussian kernel integral operator on manifolds with boundary.

翻译：我们研究了由高斯核在成对嵌入数据（从带边界的流形（$\mathbb{R}^m$ 的子流形）中采样）上评估所诱导的对称化图拉普拉斯矩阵的谱收敛性。具体而言，我们推导了离散图拉普拉斯矩阵的特征对收敛到定义良好的带边界流形上拉普拉斯-贝尔特拉米算子（包括齐次诺伊曼和狄利克雷边界条件）的本征解时的收敛速率。对于狄利克雷问题，我们推导了最近在应用中通过数值观测到的截断图拉普拉斯的收敛性，并在简单流形上提供了详细的数值研究。我们的证明方法依赖于在紧致对称积分算子上的极小-极大论证，利用再生核希尔伯特空间理论实现积分算子的谱收敛，以及近期关于带边界流形上高斯核积分算子逐点渐近结果。