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In this paper, we study the Radial Basis Function (RBF) approximation todifferential operators on smooth tensor fields defined on closed Riemanniansubmanifolds of Euclidean space, identified by randomly sampled point clouddata. {The formulation in this paper leverages a fundamental fact that thecovariant derivative on a submanifold is the projection of the directionalderivative in the ambient Euclidean space onto the tangent space of thesubmanifold. To differentiate a test function (or vector field) on thesubmanifold with respect to the Euclidean metric, the RBF interpolation isapplied to extend the function (or vector field) in the ambient Euclideanspace. When the manifolds are unknown, we develop an improved second-orderlocal SVD technique for estimating local tangent spaces on the manifold. Whenthe classical pointwise non-symmetric RBF formulation is used to solveLaplacian eigenvalue problems, we found that while accurate estimation of theleading spectra can be obtained with large enough data, such an approximationoften produces irrelevant complex-valued spectra (or pollution) as the truespectra are real-valued and positive. To avoid such an issue,} we introduce asymmetric RBF discrete approximation of the Laplacians induced by a weakformulation on appropriate Hilbert spaces. Unlike the non-symmetricapproximation, this formulation guarantees non-negative real-valued spectra andthe orthogonality of the eigenvectors. Theoretically, we establish theconvergence of the eigenpairs of both the Laplace-Beltrami operator and BochnerLaplacian {for the symmetric formulation} in the limit of large data withconvergence rates. Numerically, we provide supporting examples forapproximations of the Laplace-Beltrami operator and various vector Laplacians,including the Bochner, Hodge, and Lichnerowicz Laplacians.

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