Regular Families of Kernels for Nonlinear Approximation

TL;DR

This paper establishes conditions on kernel families that enable nonlinear approximation errors to match the best wavelet expansion orders, covering splines, radial basis functions, and more, with theoretical and computational validation.

Contribution

It introduces a unified framework of kernel conditions for optimal nonlinear approximation, including new results on radial basis function interpolation of Sobolev functions.

Findings

Conditions on kernels ensure optimal N-term approximation rates.

The framework applies to splines, radial basis functions, and cardinal functions.

Computational experiments support the theoretical results.

Abstract

This article studies sufficient conditions on families of approximating kernels which provide $N$--term approximation errors from an associated nonlinear approximation space which match the best known orders of $N$--term wavelet expansion. These conditions provide a framework which encompasses some notable approximation kernels including splines, so-called cardinal functions, and many radial basis functions such as the Gaussians and general multiquadrics. Examples of such kernels are given to justify the criteria, and some computational experiments are done to demonstrate the theoretical results. Additionally, the techniques involved allow for some new results on $N$--term interpolation of Sobolev functions via radial basis functions.