Ukrainian Mathematical Journal

, Volume 62, Issue 3, pp 452–466 | Cite as

Best approximation by ridge functions in L p -spaces

  • V. E. Maiorov

We study the approximation of the classes of functions by the manifold R n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ qp ≤ ∞, the deviation of the Sobolev class W r p from the set R n of ridge functions in the space L q (B d ) satisfies the sharp order n -r/(d-1).


Orthogonal Polynomial Orthonormal System Orthogonal System Algebraic Polynomial Multivariate Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. R. Barron, “ Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inform. Theory, 39, 930–945 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    R. A. DeVore, K. Oskolkov, and P. Petrushev, “Approximation by feed-forward neural networks,” Ann. Numer. Math., 4, 261–287 (1997).zbMATHMathSciNetGoogle Scholar
  3. 3.
    L. Devroye, L. Györfy, and G. Lugosi, A Probabilistic Theory of Pattern Recognition, Springer, New York (1996).zbMATHGoogle Scholar
  4. 4.
    A. Erdelyi, ed., Higher Transcendental Functions. Vol. 2. Bateman Manuscript Project, McGraw Hill, New York (1953).Google Scholar
  5. 5.
    Y. Gordon, V. Maiorov, M. Meyer, and S. Reisner, “On best approximation by ridge functions in the uniform norm,” Constr. Approxim., 18, 61–85 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    V. Ya. Lin and A. Pinkus, “Fundamentality of ridge functions,” J. Approxim. Theory, 75, 295–311 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    V. Ya. Lin and A. Pinkus, “Approximation of multivariate functions,” Adv. Comput. Math., World Sci. (Singapore), 257–265 (1994).Google Scholar
  8. 8.
    B. Logan and L. Shepp, “ Optimal reconstruction of functions from their projections,” Duke Math. J., 42, 645–659 (1975).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    V. Maiorov, “ On best approximation by ridge functions,” J. Approxim. Theory, 99, 68–94 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. Maiorov, “On best approximation of classes by radial functions,” J. Approxim. Theory, 120, 36–70 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    V. Maiorov, R. Meir, and J. Ratsaby, “On the approximation of functional classes equipped with a uniform measure using ridge functions,” J. Approxim. Theory, 99, 95–111 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    V. Maiorov and R. Meir, “On the near optimality of the stochastic approximation of smooth functions by neural networks,” Adv. Comput. Math., 13, 79–103 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    V. Maiorov, K. I. Oskolkov, and V. N. Temlyakov, “Gridge approximation and Radon compass,” Approxim. Theory (a Volume dedicated to Blagovest Sendov), Ed. B. Bojanov, DARBA, Sofia, 284–309 (2002).Google Scholar
  14. 14.
    Makovoz Y., “Random approximation and neural networks,” J. Approxim. Theory, 85, 98–109 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    H. N. Mhaskar, “Neural networks for optimal approximation of smooth and analytic functions,” Neural Comput., 8, 164–177 (1996).CrossRefGoogle Scholar
  16. 16.
    H. N. Mhaskar and C. A. Micchelli, “Dimension independent bounds on the degree of approximation by neural networks,” IBM J. Res. Develop., 38, 277–284 (1994).zbMATHCrossRefGoogle Scholar
  17. 17.
    K. I. Oskolkov, “Ridge approximation, Chebyshev–Fourier analysis and optimal quadrature formulas,” Proc. Steklov Inst. Math., 219, 265–280 (1997).MathSciNetGoogle Scholar
  18. 18.
    P. P. Petrushev, “Approximation by ridge functions and neural networks,” SIAM J. Math. Anal., 30, 291–300 (1998).CrossRefMathSciNetGoogle Scholar
  19. 19.
    A. Pinkus, “Approximation by ridge functions, some density problems from neutral networks,” Surface Fitting and Multiresolution Method, 2, 279–292 (1997).MathSciNetGoogle Scholar
  20. 20.
    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, New Jersey, Princeton (1971).zbMATHGoogle Scholar
  21. 21.
    V. Temlyakov, On Approximation by Ridge Functions, Preprint.Google Scholar
  22. 22.
    A. F. Timan, Theory of Approximation of Functions of the Real Variable, Macmillan Co., New York (1963).Google Scholar
  23. 23.
    H. Tribel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag Wissenschaften, Berlin (1978).Google Scholar
  24. 24.
    V. Vapnik and A. Chervonkis, “Necessary and sufficient conditions for the uniform convergence of empirical means to their expectations,” Theory Probab. Appl., 3, 532–553 (1981).Google Scholar
  25. 25.
    N. Ya. Vilenkin, Special Functions and the Theory of Representations of Groups [in Russian], Fizmatgiz, Moscow (1965).Google Scholar
  26. 26.
    B. A. Vostretsov and M. A. Kreines, “Approximation of continuous functions by superpositions of plane waves,” Sov. Math. Dokl., 2, 1320–1329 (1961).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • V. E. Maiorov
    • 1
  1. 1.TechnionHaifaIsrael

Personalised recommendations