Spectral Convergence of Graph Laplacian and Heat Kernel Reconstruction in L^∞ from Random Samples

2019/12/11 by David B. Dunson, Hau‐Tieng Wu, Dunson, David B +3 · 3 citations
Mathematics · Computer Science · #Markov Chains and Monte Carlo Methods #Topological and Geometric Data Analysis #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1912.05680

Abstract

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^∞ 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^∞ sense and providing a numerical heat kernel reconstruction from the point cloud with theoretical guarantees.

Citations

• The spectral function of an elliptic operator
• Spectral Convergence Rate of Graph Laplacian

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