Approximation complexity of additive random fields

We study approximation properties of sequences of centered random elements X d , d ∈ 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 (ε) 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 ε ∈ (0, 1). The growth of n X d (ε) as a function of ε −1 and d determines whether a sequence of corresponding approximation problems for X d , d ∈ N, is tractable or not. Different types of tractability were studied in the paper by M. A. Lifshits, A. Papageorgiou and H. Woźniakowski (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 G. Xu (2014) concerning weighted Korobov kernels.

“…On the one hand, the quantity n X d (ε) satisfies (3) for some C > 0 and s 0. On the other hand, from (1), (8), and λ X d 1 = d j=1 λ X 1,j 1…”

Section: Quasi-polynomial Tractabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Khartov

2016

“…For varying d, the homogeneous case, i.e., R k = R for all k with R not necessarily equal to a Korobov kernel, was studied in [3,4,6]. In this case, we have the curse of dimensionality since n avg (ε, d)…”

Section: Multivariate Approximation and Korobov Kernelsmentioning

confidence: 99%

Average case tractability of non-homogeneous tensor product problems

Lifshits

Papageorgiou

Woniakowski

2012

Self Cite

a b s t r a c tWe study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure ν d . Our interest is focused on measures having the structure of a nonhomogeneous tensor product, where the covariance kernel of ν d is a product of univariate kernels:We consider the normalized average error of algorithms that use finitely many evaluations of arbitrary linear functionals. The information complexity is defined as the minimal number n avg (ε, d) of such evaluations for error in the d-variate case to be at most ε. The growth of n avg (ε, d) as a function of ε −1 and d depends on the eigenvalues of the covariance operator of ν d and determines whether a problem is tractable or not. Four types of tractability are studied and for each of them we find the necessary and sufficient conditions in terms of the eigenvalues of the integral operator with kernel R k . We illustrate our results by considering approximation problems related to the product of Korobov kernels R k . Each R k is characterized by a weight g k and a smoothness r k . We assume that weights are non-increasing and smoothness parameters are nondecreasing. Furthermore they may be related; for instance g k = g(r k ) for some non-increasing function g. In particular, we show 540 M. A. Lifshits et al. / Journal of Complexity 28 (2012) 539-561 that the approximation problem is strongly polynomially tractable, i.e., n avg (ε, d) ≤ C ε −p for all d ∈ N, ε ∈ (0, 1], where C and p are independent of ε and d, iffFor other types of tractability we also show necessary and sufficient conditions in terms of the sequences g k and r k .

“…For both types of the fields, the error of linear algorithms is studied in great detail, cf. [12,11,13,18], as well as closely related small deviation behavior [6]. However, the power of standard information for them was not addressed so far, which we do here.…”

Section: Introduction: General Information Against Standard Informationmentioning

confidence: 88%

Approximation of additive random fields based on standard information: Average case and probabilistic settings

Lifshits

Zani

2015

We consider approximation problems for tensor product and additive random fields based on standard information in the average case setting. We also study the probabilistic setting of the mentioned problem for tensor products. The main question we are concerned with in this paper is "How much do we loose by considering standard information algorithms against those using general linear information?" For both types of the fields, the error of linear algorithms has been studied in great detail. However, the power of standard information for them was not addressed so far, which we do here. Our main conclusion is that in most interesting cases there is no more than a logarithmic loss in approximation error when information is being restricted to the standard one.The results are obtained by randomization techniques. *