Convergence of the Weighted Nonlocal Laplacian on Random Point Cloud (Journal of Computational Mathematics, 2021)
We analyze the convergence of the weighted nonlocal Laplacian (WNLL) on the high dimensional randomly distributed point cloud. Our analysis reveals the importance of the scaling weight, $μ \sim |P|/|S|$ with $|P|$ and $|S|$ being the number of entire and labeled data, respectively, in WNLL. The established result gives a theoretical foundation of the WNLL for high dimensional data interpolation.
Keywords: Weighted nonlocal Laplacian; Laplace-Beltrami operator; Point cloud; High-dimensional interpolation.