A simplified criterion for quasi-polynomial tractability of approximation of random elements and its applications

We study approximation properties of sequences of centered random elements X-d, d is an element of 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(Xd)(a) is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate Xd with relative 2-average error not exceeding a given threshold epsilon is an element of (0, 1). The growth of n(Xd) (a) as a function of epsilon(-1) and d determines whether a sequence of corresponding approximation problems for X-d, d is an element of N, is tractable or not. Different types of tractability were studied in the paper by Lifshits et al. (J. Complexity, 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 a simplified version. We illustrate our result by applying it to random elements corresponding to tensor products of squared exponential kernels. We also extend a recent result of Xu (2014) concerning weighted Korobov kernels. (C) 2015 Elsevier Inc. All rights reserved.