Nonlinear Approximation Using Gaussian Kernels

TL;DR

This paper introduces a new nonlinear approximation algorithm using Gaussian kernels with varying tension parameters, achieving approximation rates comparable to spline and wavelet methods, and adaptable to the smoothness of the target function.

Contribution

It develops a novel algorithm for Gaussian approximation with spatially varying tension parameters, extending nonlinear approximation theory to Gaussian kernels.

Findings

Algorithm achieves near-optimal approximation rates

Approximation rates depend on the smoothness of the target function

Method extends nonlinear approximation results to Gaussian kernels

Abstract

It is well-known that non-linear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore and Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.