Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant

Abstract

The purpose of this paper is to establish that for any compact, connected C^{\infty} Riemannian manifold there exists a robust family of kernels of increasing smoothness that are well suited for interpolation. They generate Lagrange functions that are uniformly bounded and decay away from their center at an exponential rate. An immediate corollary is that the corresponding Lebesgue constant will be uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The analysis needed for these results was inspired by some fundamental work of Matveev where the Sobolev decay of Lagrange functions associated with certain kernels on Ω\subset R^d was obtained. With a bit more work, one establishes the following: Lebesgue constants associated with surface splines and Sobolev splines are uniformly bounded on R^d provided the data sites Ξare quasi-uniformly distributed. The non-Euclidean case is more involved as the geometry of the underlying surface comes into play. In addition to establishing bounded Lebesgue constants in this setting, a "zeros lemma" for compact Riemannian manifolds is established.

33 pages, 2 figures, new title, accepted for publication in SIAM J. on Math. Anal

Cited by in corpus (20)

• Reproducing Kernel Banach Spaces with the l1 Norm
• Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation
• Kernel Approximation on Manifolds II: The $L_{\infty }$-norm of the $L_2$-projector
• Localized bases for kernel spaces on the unit sphere
• Cardinal Interpolation With General Multiquadrics
• Massively scalable Sinkhorn distances via the Nyström method
• Error Bounds for a Least Squares Meshless Finite Difference Method on Closed Manifolds
• Persistence of Excitation in Reproducing Kernel Hilbert Spaces, Positive Limit Sets, and Smooth Manifolds
• Surface Spline Approximation on SO(3)
• An inverse theorem for compact Lipschitz regions in $R^d$ using localized kernel bases
• Approximations of the Reproducing Kernel Hilbert Space (RKHS) Embedding Method over Manifolds
• Kernel approximation on algebraic varieties
• The Penalized Lebesgue Constant for Surface Spline Interpolation
• Kernel Interpolation of High Dimensional Scattered Data
• Smoothness Estimation for Whittle-Matérn Processes on Closed Riemannian Manifolds
• Deep Networks Provably Classify Data on Curves
• Multi-task Learning in Vector-valued Reproducing Kernel Banach Spaces with the ${\ell }^1$ Norm
• Kernel Based Quadrature on Spheres and Other Homogeneous Spaces
• Vector-valued Reproducing Kernel Banach Spaces with Group Lasso Norms
• Optimal order Jackson type inequality for scaled Shepard approximation