Nonlinear Function Learning Using Radial Basis Function Networks: Convergence and Rates

  • Adam Krzyżak
  • Dominik Schäfer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5097)


We apply normalized RBF networks to the problem of learning nonlinear regression functions. The parameters of the networks are learned by empirical risk minimization and complexity regularization. We study convergence of the RBF networks for various radial kernels as the number of training samples increases. The rates of convergence are also examined.


Normalized radial basis function networks convergence rates of convergence 


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  1. 1.
    Anthony, M., Bartlett, P.L.: Neural Network Learning: Theoretical Foundations. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  2. 2.
    Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. on Information Theory 39, 930–945 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Barron, A.R.: Approximation and estimation bounds for artificial neural networks. Machine Learning 14, 115–133 (1994)zbMATHGoogle Scholar
  4. 4.
    Cybenko, G.: Approximations by superpositions of sigmoidal functions. Mathematics of Control, Signals, and Systems 2, 303–314 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Devroye, L., Györfi, L., Lugosi, G.: Probabilistic Theory of Pattern Recognition. Springer, New York (1996)zbMATHGoogle Scholar
  6. 6.
    Duchon, J.: Sur l’erreur d’interpolation des fonctions de plusieurs variables par les Dm-splines. RAIRO Anal. Numér. 12(4), 325–334 (1978)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Faragó, A., Lugosi, G.: Strong universal consistency of neural network classifiers. IEEE Trans. on Information Theory 39, 1146–1151 (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    Girosi, F.: Regularization theory, radial basis functions and networks. In: Cherkassky, V., Friedman, J.H., Wechsler, H. (eds.) From Statistics to Neural Networks. Theory and Pattern recognition Applications, pp. 166–187. Springer, Berlin (1992)Google Scholar
  9. 9.
    Girosi, F., Anzellotti, G.: Rates of convergence for radial basis functions and neural networks. In: Mammone, R.J. (ed.) Artificial Neural Networks for Speech and Vision, pp. 97–113. Chapman and Hall, London (1993)Google Scholar
  10. 10.
    Girosi, F., Jones, M., Poggio, T.: Regularization theory and neural network architectures. Neural Computation 7, 219–267 (1995)CrossRefGoogle Scholar
  11. 11.
    Grenander, U.: Abstract Inference. Wiley, New York (1981)zbMATHGoogle Scholar
  12. 12.
    Györfi, L., Kohler, M., Krzyżak, A., Walk, H.: A DistributionFree Theory of Nonparametric Regression. Springer, New York (2002)Google Scholar
  13. 13.
    Haussler, D.: Decision theoretic generalizations of the PAC model for neural net and other learning applications. Information and Computation 100, 78–150 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Hornik, K., Stinchocombe, S., White, H.: Multilayer feed-forward networks are universal approximators. Neural Networks 2, 359–366 (1989)CrossRefGoogle Scholar
  15. 15.
    Kolmogorov, A.N., Tihomirov, V.M.: ε-entropy and ε-capacity of sets in function spaces. Translations of the American Mathematical Society 17, 277–364 (1961)MathSciNetGoogle Scholar
  16. 16.
    Krzyżak, A., Linder, T., Lugosi, G.: Nonparametric estimation and classification using radial basis function nets and empirical risk minimization. IEEE Trans. Neural Networks 7(2), 475–487 (1996)CrossRefGoogle Scholar
  17. 17.
    Krzyżak, A., Linder, T.: Radial basis function networks and complexity regularization in function learning. IEEE Trans. Neural Networks 9(2), 247–256 (1998)CrossRefGoogle Scholar
  18. 18.
    Krzyżak, A., Niemann, H.: Convergence and rates of convergence of radial basis functions networks in function learning. Nonlinear Analysis 47, 281–292 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Krzyżak, A., Schäfer, D.: Nonparametric regression estimation by normalized radial basis function networks. IEEE Transactions on Information Theory 51(3), 1003–1010 (2005)CrossRefGoogle Scholar
  20. 20.
    Lugosi, G., Zeger, K.: Nonparametric estimation via empirical risk minimization. IEEE Trans. on Information Theory 41, 677–687 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Moody, J., Darken, J.: Fast learning in networks of locally-tuned processing units. Neural Computation 1, 281–294 (1989)CrossRefGoogle Scholar
  22. 22.
    Park, J., Sandberg, I.W.: Universal approximation using Radial-Basis-Function networks. Neural Computation 3, 246–257 (1991)CrossRefGoogle Scholar
  23. 23.
    Park, J., Sandberg, I.W.: Approximation and Radial-Basis-Function networks. Neural Computation 5, 305–316 (1993)CrossRefGoogle Scholar
  24. 24.
    Pollard, D.: Convergence of Stochastic Processes. Springer, New York (1984)zbMATHGoogle Scholar
  25. 25.
    Rissanen, J.: A universal prior for integers and estimation by minimum description length. Annals of Statistics 11, 416–431 (1983)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Ripley, B.D.: Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  27. 27.
    Shorten, R., Murray-Smith, R.: Side effects of normalising radial basis function networks. International Journal of Neural Systems 7, 167–179 (1996)CrossRefGoogle Scholar
  28. 28.
    Specht, D.F.: Probabilistic neural networks. Neural Networks 3, 109–118 (1990)CrossRefGoogle Scholar
  29. 29.
    Vapnik, V.N.: Estimation of Dependences Based on Empirical Data, 2nd edn. Springer, New York (1999)Google Scholar
  30. 30.
    Xu, L., Krzyżak, A., Yuille, A.L.: On radial basis function nets and kernel regression: approximation ability, convergence rate and receptive field size. Neural Networks 7, 609–628 (1994)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Adam Krzyżak
    • 1
    • 3
  • Dominik Schäfer
    • 2
  1. 1.Department of Computer Science and Software EngineeringConcordia UniversityMontrealCanada
  2. 2.Department of MathematcsStuttgart UniversityStuttgartGermany
  3. 3.Institute of Control EngineeringTechnical University of SzczecinSzczecinPoland

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