Tractability of the function approximation problem in terms of the kernel's shape and scale parameters

TL;DR

This paper investigates the conditions under which function approximation in high-dimensional Hilbert spaces with a generalized kernel is computationally feasible, focusing on how kernel shape and scale parameters influence tractability.

Contribution

It provides new sufficient conditions on kernel parameters ensuring polynomial tractability for high-dimensional function approximation problems.

Findings

Polynomial tractability depends on the product of scale and shape parameters.

Eigenvalue bounds determine the exponent of strong polynomial tractability.

Generalizes known results for anisotropic Gaussian kernels.

Abstract

This article studies the problem of approximating functions belonging to a Hilbert space $\mathcal H_d$ with a reproducing kernel of the form $$\tilde K_d(\boldsymbol x,\boldsymbol t):=\prod_{\ell=1}^d \left(1-\alpha_\ell^2+\alpha_\ell^2K_{\gamma_\ell}(x_\ell,t_\ell)\right)\ \ \ \mbox{for all} \ \ \ \boldsymbol x,\boldsymbol t\in\mathbb R^d.$$ The $\alpha_\ell\in[0,1]$ are scale parameters, and the $\gamma_\ell>0$ are sometimes called shape parameters. The reproducing kernel $K_{\gamma}$ corresponds to some Hilbert space of functions defined on $\mathbb R$. The kernel $\tilde K_d$ generalizes the anisotropic Gaussian reproducing kernel, whose tractability properties have been established in the literature. We present sufficient conditions on $\{\alpha_\ell \gamma_\ell\}_{\ell=1}^{\infty}$ under which polynomial tractability holds for function approximation problems on $\mathcal H_d$. The exponent of strong polynomial tractability arises from bounds on the eigenvalues of a positive definite linear operator.