Rate of Convergence and Tractability of the Radial Function Approximation Problem

Abstract

This article studies the problem of approximating functions belonging to a Hilbert space $H_d$ with an isotropic or anisotropic Gaussian reproducing kernel, $$ K_d(\bx,\bt) = \exp\left(-\sum_{\ell=1}^dγ_\ell^2(x_\ell-t_\ell)^2\right) \ \ \ \mbox{for all}\ \ \bx,\bt\in\reals^d. $$ 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 $d$.

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