  In the manifold setting, we provide a series of spectral convergence results
quantifying how the eigenvectors and eigenvalues of the graph Laplacian
converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator
in the $L^\infty$ sense. The convergence rate is also provided. Based on these
results, convergence of the proposed heat kernel approximation algorithm, as
well as the convergence rate, to the exact heat kernel is guaranteed. To our
knowledge, this is the first work exploring the spectral convergence in the
$L^\infty$ sense and providing a numerical heat kernel reconstruction from the
point cloud with theoretical guarantees.
