On Approximation Complexity in Average Case Setting for Tensor Degrees of Random Processes

We consider a random field with a zero mean and a continuous covariance function that is a d -tensor degree of a second-order random process. The average case approximation complexity of a given random field is defined as the minimal number of evaluations of linear functionals needed to approximate...

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Veröffentlicht in:Vestnik, St. Petersburg University. Mathematics Jg. 58; H. 1; S. 59 - 70
Hauptverfasser: Pyatkin, K. A., Khartov, A. A.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Moscow Pleiades Publishing 01.03.2025
Springer Nature B.V
Schlagworte:
Analysis
Approximation
Complexity
Eigenvalues
Fields (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
Probability
Random processes
Random variables
Tensors
random fields
average case setting
high dimension
tractability
tensor degree
approximation complexity
ISSN:1063-4541, 1934-7855
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Zusammenfassung:We consider a random field with a zero mean and a continuous covariance function that is a d -tensor degree of a second-order random process. The average case approximation complexity of a given random field is defined as the minimal number of evaluations of linear functionals needed to approximate the field with a relative r.m.s. error not exceeding a given threshold ε. This paper gives an upper estimate for that is always valid (without any criteria) for any ε and d . The logarithm of this estimate is in well agreement with the asymptotics obtained by us as d → ∞ with a threshold ε = ε d , which can rather quickly converge to zero as d → ∞. The estimate and the asymptotics complement and generalize the results obtained by Lifshits and Tulyakova as well as by Kravchenko and Khartov in this direction.
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ISSN:1063-4541
1934-7855
DOI:10.1134/S1063454125700074