Simplified criterion of quasi-polynomial tractability and its applications

TL;DR

This paper revises and simplifies the criterion for quasi-polynomial tractability in high-dimensional approximation problems, applying it to tensor product kernels and extending previous results on weighted Korobov kernels.

Contribution

It provides a simplified criterion for quasi-polynomial tractability and demonstrates its application to tensor product kernels and weighted Korobov kernels.

Findings

Simplified the criterion for quasi-polynomial tractability.

Applied the criterion to tensor products of squared exponential kernels.

Extended results to weighted Korobov kernels.

Abstract

We study approximation properties of sequences of centered random elements $X_d$, $d\in\mathbb{N}$, with values in separable Hilbert spaces. We focus on sequences of tensor product-type random elements, which have covariance operators of corresponding tensor product form. The average case approximation complexity $n^{X_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate $X_d$ with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. The growth of $n^{X_d}(\varepsilon)$ as a function of $\varepsilon^{-1}$ and $d$ determines whether a sequence of corresponding approximation problems for $X_d$, $d\in\mathbb{N}$, is tractable or not. Different types of tractability were studied in the paper by M. A. Lifshits, A. Papageorgiou and H. Wo\'zniakowski (2012), where for each type the necessary and sufficient conditions were found in terms of the eigenvalues of the marginal covariance operators. We revise the criterion of quasi-polynomial tractability and provide its simplified version. We illustrate our result by applying to random elements corresponding to tensor products of squared exponential kernels. Also we extend recent result of G. Xu (2014) concerning weighted Korobov kernels.