Asymptotic analysis of average case approximation complexity of Hilbert space valued random elements

We study approximation properties of sequences of centered random elements Xd, d∈N, with values in separable Hilbert spaces. We focus on sequences of tensor product-type and, in particular, degree-type random elements, which have covariance operators of a corresponding tensor form. The average case...

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Bibliographic Details
Published in:Journal of Complexity Vol. 31; no. 6; pp. 835 - 866
Main Author: Khartov, A.A.
Format: Journal Article
Language:English
Published: Elsevier Inc 01.12.2015
Subjects:
Approximation
Asymptotic analysis
Average case approximation complexity
Multivariate problems
Tensor product-type random elements
Tractability
Tractability
Multivariate problems
Tensor product-type random elements
Approximation
Asymptotic analysis
Average case approximation complexity
ISSN:0885-064X, 1090-2708
Online Access:Get full text
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Summary:We study approximation properties of sequences of centered random elements Xd, d∈N, with values in separable Hilbert spaces. We focus on sequences of tensor product-type and, in particular, degree-type random elements, which have covariance operators of a corresponding tensor form. The average case approximation complexity nXd(ε) is defined as the minimal number of continuous linear functionals that is needed to approximate Xd with a relative 2-average error not exceeding a given threshold ε∈(0,1). In the paper we investigate nXd(ε) for arbitrary fixed ε∈(0,1) and d→∞. Namely, we find criteria of (un) boundedness for nXd(ε) on d and of tending nXd(ε)→∞, d→∞, for any fixed ε∈(0,1). In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptoticslnnXd(ε)=ad+q(ε)bd+o(bd),d→∞, at continuity points of a non-decreasing function q:(0,1)→R. Here (ad)d∈N is a sequence and (bd)d∈N is a positive sequence such that bd→∞, d→∞. Under rather weak assumptions, we show that for tensor product-type random elements only special quantiles of self-decomposable or, in particular, stable (for tensor degrees) probability distributions appear as functions q in the asymptotics. We apply our results to the tensor products of the Euler integrated processes with a given variation of smoothness parameters and to the tensor degrees of random elements with regularly varying eigenvalues of covariance operator.
ISSN:0885-064X
1090-2708
DOI:10.1016/j.jco.2015.06.004