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N-Widths and ε-Dimensions for High-Dimensional Approximations

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Abstract

In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths d n (W,H γ), and ε-dimensions n ε (W,H γ) of periodic d-variate function classes W with anisotropic smoothness, where d may be large. We are interested in finding the accurate dependence of d n (W,H γ) and n ε (W,H γ) as a function of two variables n, d and ε, d, respectively. Recall that n, the dimension of the approximating subspace, is the main parameter in the study of convergence rates with respect to n going to infinity. However, the parameter d may seriously affect this rate when d is large. We construct linear approximations of functions from W by trigonometric polynomials with frequencies from hyperbolic crosses and prove upper bounds for the error measured in isotropic Sobolev spaces H γ. Furthermore, in order to show the optimality of the proposed approximation, we prove upper and lower bounds of the corresponding n-widths d n (W,H γ) and ε-dimensions n ε (W,H γ). Some of the received results imply that the curse of dimensionality can be broken in some relevant situations.

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Acknowledgements

The work of the first named author was supported by Grant 102.01-2012.15 of the National Foundation for Development of Science and Technology (Vietnam). The both authors would like to thank the Hausdorff Research Institute for Mathematics (HIM) and the organizers of the HIM Trimester Program “Analysis and Numerics for High Dimensional Problems”, where this paper was initiated, for providing a fruitful research environment and additional financial support. Last but not least, the authors would like to thank the referees for a critical reading of the manuscript and for several valuable suggestions which helped to improve its presentation.

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Authors and Affiliations

  1. Information Technology Institute, Vietnam National University, Hanoi, 144 Xuan Thuy, Hanoi, Vietnam

    Dinh Dũng

  2. Hausdorff-Center for Mathematics and Institute for Numerical Simulation, 53115, Bonn, Germany

    Tino Ullrich

Authors
  1. Dinh Dũng
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  2. Tino Ullrich
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Correspondence to Dinh Dũng.

Additional information

Communicated by Wolfgang Dahmen.

Dedicated to the memory of Professor S.M. Nikol’skij.

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Dũng, D., Ullrich, T. N-Widths and ε-Dimensions for High-Dimensional Approximations. Found Comput Math 13, 965–1003 (2013). https://doi.org/10.1007/s10208-013-9149-9

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  • DOI: https://doi.org/10.1007/s10208-013-9149-9

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