DOI QR코드

DOI QR Code

THE SIMULTANEOUS APPROXIMATION ORDER BY NEURAL NETWORKS WITH A SQUASHING FUNCTION

  • Published : 2009.07.31

Abstract

In this paper, we study the simultaneous approximation to functions in $C^m$[0, 1] by neural networks with a squashing function and the complexity related to the simultaneous approximation using a Bernstein polynomial and the modulus of continuity. Our proofs are constructive.

Keywords

References

  1. R. M. Burton and H. G. Dehling, Universal approximation in p-mean by neural networks, Neural Networks 11 (1998), 661–667 https://doi.org/10.1016/S0893-6080(98)00009-4
  2. P. Cardaliaguet and G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Networks 5 (1992), 207–220 https://doi.org/10.1016/S0893-6080(05)80020-6
  3. R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer Verlag, Heidelberg, 1993
  4. A. R. Gallant and H. White, On learning the derivatives of an unknown mapping with multilayer feedforward networks, Lett. Math. Phys. 5 (1992), 129–138 https://doi.org/10.1016/S0893-6080(05)80011-5
  5. B. Gao and Y. Xu, Univariant approximation by superpositions of a sigmoidal function, J. Math. Anal. Appl. 178 (1993), no. 1, 221–226 https://doi.org/10.1006/jmaa.1993.1302
  6. Y. Ito, Simultaneous Lp-approximations of polynomials and dervatives on the whole space, Arti. Neural Networks Conf. (1999), 587–592
  7. G. Lewicki and G. Marino, Approximation of functions of finite variation by superpositions of a sigmoidal function, Appl. Math. Lett. 17 (2004), no. 10, 1147–1152 https://doi.org/10.1016/j.aml.2003.11.006
  8. F. Li and Z. Xu, The essential order of simultaneous approximation for neural networks, Appl. Math. Comput. 194 (2007), no. 1, 120–127 https://doi.org/10.1016/j.amc.2007.04.017
  9. X. Li, Simultaneous approximation of a multivariate functions and their derivatives by neural networks with one hidden layer, Neurocomputing 12 (1996), 327–343 https://doi.org/10.1016/0925-2312(95)00070-4
  10. G. G. Lorentz, Bernstein Polynomials, Chelsea, Engelwood Cliffs, 1986
  11. B. Malakooti and Y. Q. Zhou, Approximation polynomial functions by feedforward artificial neural networks : capacity analysis and design, Appl. Math. and Comp. 90 (1998), 27–51 https://doi.org/10.1016/S0096-3003(96)00338-4
  12. M. V. Medvedeva, On sigmoidal functions, Moscow Univ. Math. Bull. 53 (1998), no. 1, 16–19
  13. H. N. Mhaskar and N. Hahm, Neural networks for functional approximation and system identification, Neural Computation 9 (1997), no. 1, 143–159 https://doi.org/10.1162/neco.1997.9.1.143
  14. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Parallel Distributed Processing : explorations in the microstructure of cognition, MIT Press, Massachusetts, 1986