Rate of Convergence and Tractability of the Radial Function Approximation Problem

TL;DR

This paper investigates the convergence rates and computational tractability of approximating functions in high-dimensional Hilbert spaces using Gaussian kernels with isotropic and anisotropic parameters, focusing on moderate to large dimensions.

Contribution

It provides new insights into the convergence behavior and tractability of Gaussian kernel approximation problems in high dimensions, considering both isotropic and anisotropic cases.

Findings

Derived convergence rate bounds for Gaussian kernel approximation.

Analyzed the impact of shape parameters on tractability.

Identified conditions for polynomial and exponential tractability.

Abstract

This article studies the problem of approximating functions belonging to a Hilbert space $H_d$ with an isotropic or anisotropic Gaussian reproducing kernel, $$ K_d(\bx,\bt) = \exp\left(-\sum_{\ell=1}^d\gamma_\ell^2(x_\ell-t_\ell)^2\right) \ \ \ \mbox{for all}\ \ \bx,\bt\in\reals^d. $$ The isotropic case corresponds to using the same shape parameters for all coordinates, namely $\gamma_\ell=\gamma>0$ for all $\ell$, whereas the anisotropic case corresponds to varying shape parameters $\gamma_\ell$. We are especially interested in moderate to large $d$.