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.

翻译：在多重环境下,我们提供一系列光谱趋同结果,以量化Laplacian图的元素元体和元值如何与Laplace-Beltrami操作员的元件和元值相融合。%% 也提供了聚合率。根据这些结果,拟议热内核近似算法的趋同率以及聚合率得到保证,精确的热内核得到保证。 据我们所知,这是首次探索L ⁇ infty$意义的光谱趋同,并提供理论保证,从点云中进行数字热内核重建。