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

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.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1012.260

Topics: Mathematics - Numerical Analysis
Year: 2014
OAI identifier: oai:arXiv.org:1411.0790

Suggested articles

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.