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Kernel Networks with Fixed and Variable Widths

  • Věra Kůrková
  • Paul C. Kainen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6593)

Abstract

The role of width in kernel models and radial-basis function networks is investigated with a special emphasis on the Gaussian case. Quantitative bounds are given on kernel-based regularization showing the effect of changing the width. These bounds are shown to be d-th powers of width ratios, and so they are exponential in the dimension of input data.

Keywords

Kernel models Gaussian kernel networks Minimization of error functionals Regularization 

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References

  1. 1.
    Fine, T.L.: Feedforward Neural Network Methodology. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  2. 2.
    Kecman, V.: Learning and Soft Computing. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  3. 3.
    Park, J., Sandberg, I.: Universal approximation using radial–basis–function networks. Neural Computation 3, 246–257 (1991)CrossRefGoogle Scholar
  4. 4.
    Park, J., Sandberg, I.: Approximation and radial basis function networks. Neural Computation 5, 305–316 (1993)CrossRefGoogle Scholar
  5. 5.
    Mhaskar, H.N.: Versatile Gaussian networks. In: Proceedings of IEEE Workshop of Nonlinear Image Processing, pp. 70–73 (1995)Google Scholar
  6. 6.
    Kainen, P.C., Kůrková, V., Sanguineti, M.: Complexity of Gaussian radial basis networks approximating smooth functions. J. of Complexity 25, 63–74 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cucker, F., Smale, S.: On the mathematical foundations of learning. Bulletin of AMS 39, 1–49 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Poggio, T., Smale, S.: The mathematics of learning: dealing with data. Notices of AMS 50, 537–544 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kůrková, V.: Neural network learning as an inverse problem. Logic Journal of IGPL 13, 551–559 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gribonval, R., Vandergheynst, P.: On the exponential convergence of matching pursuits in quasi-incoherent dictionaries. IEEE Trans. on Information Theory 52, 255–261 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Aronszajn, N.: Theory of reproducing kernels. Transactions of AMS 68, 337–404 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Strichartz, R.: A Guide to Distribution Theory and Fourier Transforms. World Scientific, NJ (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Loustau, S.: Aggregation of SVM classifiers using Sobolev spaces. Journal of Machine Learning Research 9, 1559–1582 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Girosi, F.: An equivalence between sparse approximation and support vector machines. Neural Computation (AI memo 1606) 10, 1455–1480 (1998)CrossRefGoogle Scholar
  15. 15.
    Girosi, F., Poggio, T.: Regularization algorithms for learning that are equivalent to multilayer networks. Science 247(4945), 978–982 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Girosi, F., Jones, M., Poggio, T.: Regularization theory and neural networks architectures. Neural Computation 7, 219–269 (1995)CrossRefGoogle Scholar
  17. 17.
    Kůrková, V.: Learning from data as an inverse problem in reproducing kernel Hilbert spaces. Inverse Problems in Science and Engineering (2010) (submitted)Google Scholar
  18. 18.
    Wahba, G.: Splines Models for Observational Data. SIAM, Philadelphia (1990)CrossRefzbMATHGoogle Scholar
  19. 19.
    Friedman, A.: Modern Analysis. Dover, New York (1982)zbMATHGoogle Scholar
  20. 20.
    Kůrková, V., Neruda, R.: Uniqueness of functional representations by Gaussian basis function networks. In: Proceedings of ICANN 1994, pp. 471–474. Springer, London (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Věra Kůrková
    • 1
  • Paul C. Kainen
    • 2
  1. 1.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPrague 8Czech Republic
  2. 2.Department of MathematicsGeorgetown UniversityWashingtonUSA

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