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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| Vydáno v: | Journal of Complexity Ročník 31; číslo 6; s. 835 - 866 |
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| Jazyk: | angličtina |
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01.12.2015
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| ISSN: | 0885-064X, 1090-2708 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Khartov, A.A. |
| Author_xml | – sequence: 1 givenname: A.A. surname: Khartov fullname: Khartov, A.A. email: alexeykhartov@gmail.com organization: Chebyshev Laboratory, Department of Mathematics and Mechanics, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178, Russia |
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| Cites_doi | 10.1007/BF00533089 10.1090/S0002-9947-1952-0048726-0 10.1214/aoms/1177693075 10.1007/PL00000445 10.1214/aop/1055425782 10.1016/j.jco.2012.05.003 10.1214/aoms/1177693320 10.1016/0021-9045(86)90043-2 10.1214/aop/1176995474 10.1112/jlms/s2-33.3.395 10.1090/S0002-9947-07-04233-X 10.1016/j.jco.2007.11.002 10.1112/blms/13.1.23 10.1007/s00440-004-0337-z 10.1007/s11202-008-0105-x 10.1007/BF00534763 |
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| Keywords | Tractability Multivariate problems Tensor product-type random elements Approximation Asymptotic analysis Average case approximation complexity |
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| SubjectTerms | Approximation Asymptotic analysis Average case approximation complexity Multivariate problems Tensor product-type random elements Tractability |
| Title | Asymptotic analysis of average case approximation complexity of Hilbert space valued random elements |
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