Approximation complexity of tensor product-type random fields with heavy spectrum

Mathematics

Abstract

We consider a sequence of Gaussian tensor product-type random fields
, where Open image in new window and Open image in new window are all positive eigenvalues and eigenfunctions of the covariance operator of the process X 1, Open image in new window are standard Gaussian random variables, and Open image in new window is a subset of positive integers. For each d ∈ ℕ, the sample paths of X d almost surely belong to L 2([0, 1] d ) with norm ∥·∥2,d . The tuples Open image in new window , are the eigenpairs of the covariance operator of X d . We approximate the random fields X d , dOpen image in new window , by the finite sums X d (n) corresponding to the n maximal eigenvalues λ k , Open image in new window .
We investigate the logarithmic asymptotics of the average approximation complexity
$$n_d^{pr} (\varepsilon ,\delta ): = \min \left\{ {n \in \mathbb{N}:\mathbb{P}(\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 > \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 ) \leqslant \delta } \right\},$$
and the probabilistic approximation complexity
$$n_d^{avg} (\varepsilon ): = \min \left\{ {n \in \mathbb{N}:\mathbb{E}\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 \leqslant \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 } \right\}$$
, as the parametric dimension d → ∞ the error threshold ɛ ∈ (0, 1) is fixed, and the confidence level δ = δ(d, ɛ) is allowed to approach zero. Supplementing recent results of M.A. Lifshits and E.V. Tulyakova, we consider the case where the sequence Open image in new window decreases regularly and sufficiently slowly to zero, which has not been previously studied.

Keywords

Random Field Sample Path Covariance Operator Approximation Complexity Error Threshold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Adler and J. Taylor, Random Fields and Geometry (Springer, New York, 2007).MATHGoogle Scholar
  2. 2.
    K. Ritter, “Average-case analysis of numerical problems,” in Lecture Notes in Math. (Springer, Berlin, 2000), No. 1733.Google Scholar
  3. 3.
    E. Novak, I. H. Sloan, J. F. Traub, and H. Wózniakowski, Essays on the Complexity of Continuous Problems (European Mathematical Society Publishing House, Zürich, 2009).MATHCrossRefGoogle Scholar
  4. 4.
    M. A. Lifshits and E. V. Tulyakova, “Curse of dimentionality in approximation of random fields,” Probab. Math. Statist. 26(1), 97–112 (2006).MathSciNetMATHGoogle Scholar
  5. 5.
    G. Christoph and W. Wolf, “Convergence Theorems with a Stable Limit Law,” in Math. Research 70 (Akad. Verlag, Berlin, 1992).Google Scholar

Copyright information

© Allerton Press, Inc. 2013

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations