Radial basis approximation of tensor fields on manifolds: From operator estimation to manifold learning

We study the Radial Basis Function (RBF) approximation to differential operators on smooth tensor fields defined on closed Riemannian submanifolds of Euclidean space, identified by randomly sampled point cloud data. We establish the spectral convergence for the classical pointwise RBF discrete non-symmetric approximation of Laplacians. Numerically, we found that this formulation produces a very accurate estimation of leading spectra with large enough data, which leads to a computationally expensive task of solving an eigenvalue problem of not only large but also dense, non-symmetric matrix. However, when the size data is small and/or when the local tangent plane of the unknown manifold is poorly estimated, the accuracy deteriorates. Particularly, this formulation produces irrelevant eigenvalues, in the sense that they are not approximating any of the underlying Laplacian spectra. While these findings suggest that the RBF pointwise formulation may not be reliable to approximate Laplacians for manifold learning, it is still an effective method to approximate general differential operators on smooth manifolds for other applications, including solving PDEs and supervised learning. When the manifolds are unknown, the error bound of the pointwise operator estimation depends on the accuracy of the approximate local tangent spaces. To improve this approximation accuracy, we develop a second-order local SVD technique for estimating local tangent spaces on the manifold that offsets the errors induced by the curvature in the classical first-order local SVD technique. For manifold learning, we introduce a symmetric RBF discrete approximation of the Laplacians induced by a weak formulation on appropriate Hilbert spaces. We establish the convergence of the eigenpairs of both the Laplace-Beltrami operator and Bochner Laplacian in the limit of large data, and provide supporting numerical examples.