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 XdXd, d∈Nd∈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 nXd(ε)nXd(ε) is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate XdXd with relative 22-average error not exceeding a given threshold ε∈(0,1)ε∈(0,1). The growth of nXd(ε)nXd(ε) as a function of ε−1ε−1 and dd determines whether a sequence of corresponding approximation problems for XdXd, d∈Nd∈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.