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:具体地说,我们从一组嵌入数据的高斯内核样本中,从一组带有边界的方块中抽取出,一个离散的图形-拉普拉西安基质基质的分元体的光谱集合率,推算出一个在有边界的方块(包括单一的纽曼和迪里赫莱特边界条件)上明确界定的拉皮尔-贝特拉米操作员的纤维溶液上,由一组内嵌数据(从带有边界的方块中抽取)所导出的一个对称的拉普拉西安基质图的相光谱聚合率。关于Drichlet问题,我们推算出最近在应用中用数字观测到的 \emph{ trunced Grap Laplecian} 的汇合率,并提供了对简单方块的详细数字调查。我们的证据方法依赖于对一个紧凑和对称综合操作员的微轴轴引力,利用RKHS理论来使整体操作员的光谱融合,以及一个高斯内核内核综合操作员与边界的断结果。