Spectral convergence of symmetrized graph Laplacian on manifolds with boundary

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 truncated graph Laplacian, whose convergence was recently observed numerically in applications, and provide a detailed numerical investigation on simple manifolds. Our method of proof relies on a min-max argument over a compact and symmetric integral operator, leveraging the RKHS theory for spectral convergence of integral operators and a recent asymptotic result of a Gaussian kernel integral operator on manifolds with boundary.