Univariate sigmoidal neural network approximation

Journal of Computational Analysis and Applications

Volumn 14, Issue 4, 2012, Pages 659-690

  Anastassiou, George A ^a  

^a University of Memphis   (United States)

Author keywords

Complex approximation; Modulus of continuiuty; Neural network approximation; Quasi interpolation operator; Sigmoidal function

Indexed keywords

EID: 84856712031     PISSN: 15211398     EISSN: 15729206     Source Type: Journal    
DOI: None     Document Type: Article

References (25)

1. G.A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, J. Math. Anal. Appli. 212 (1997), 237-262.
2. G.A. Anastassiou, Quantitative Approximations, Chapman&Hall/CRC, Boca Raton, New York, 2001.
3. G.A. Anastassiou, Basic Inequalities, Revisited, Mathematica Balkanica, New Series, Vol. 24, Fasc. 1-2 (2010), 59-84.
4. A.R. Barron, Universal approximation bounds for superpositions of a sig- moidal function, IEEE Trans. Inform. Theory, 39 (1993), 930-945.
5. F. Brauer and C. Castillo-Chavez, Mathematical models in population biol- ogy and epidemiology, Springer-Verlag, New York, pp. 8-9, 2001.
6. F.L. Cao, T.F. Xie and Z.B. Xu, The estimate for approximation error of neural networks: a constructive approach, Neurocomputing, 71 (2008), 626-630.
7. T.P. Chen and H. Chen, Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its applications to a dynamic system, IEEE Trans. Neural Networks, 6 (1995), 911-917.
8. Z. Chen and F. Cao, The approximation operators with sigmoidal functions, Computers and Mathematics with Applications, 58 (2009), 758-765.
9. C.K. Chui and X. Li, Approximation by ridge functions and neural networks with one hidden layer, J. Approx. Theory, 70 (1992), 131-141.
10. G. Cybenko, Approximation by superpositions of sigmoidal function, Math. of Control Signals and System, 2 (1989), 303-314.
11. S. Ferrari and R.F. Stengel, Smooth function approximation using neural networks, IEEE Trans. Neural Networks, 16 (2005), 24-38.
12. K.I. Funahashi, On the approximate realization of continuous mappings by neural networks, Neural Networks, 2 (1989), 183-192.
13. N. Hahm and B.I. Hong, An approximation by neural networks with a fixed weight, Computers & Math. with Appli., 47 (2004), 1897-1903.
14. K. Hornik, M. Stinchombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks, 2 (1989), 359-366.
15. K. Hornik, M. Stinchombe and H. White, Universal approximation of an unknown mapping and its derivatives using multilayer feedforward net- works, Neural Networks, 3 (1990), 551-560.
16. N. Hritonenko and Y. Yatsenko, Mathematical modeling in economics, ecol- ogy and the environment, Reprint, Science Press, Beijing, pp. 92-93, 2006.
17. M. Leshno, V.Y. Lin, A. Pinks and S. Schocken, Multilayer feedforward networks with a nonpolynomial activation function can approximate any function, Neural Networks, 6 (1993), 861-867.
18. G.G. Lorentz, Approximation of Functions, Rinehart and Winston, New York, 1966.
19. d) by neural and mixture networks, IEEE Trans. Neural Networks, 9 (1998), 969-978.
20. Y. Makovoz, Uniform approximation by neural networks, J. Approx. The- ory, 95 (1998), 215-228.
21. H.N. Mhaskar and C.A. Micchelli, Approximation by superposition of a sigmoidal function, Adv. Applied Math., 13 (1992), 350-373.
22. H.N. Mhaskar and C.A. Micchelli, Degree of approximation by neural net- works with a single hidden layer, Adv. Applied Math., 16 (1995), 151-183.
23. S. Suzuki, Constructive function approximation by three-layer artificial neu- ral networks, Neural Networks, 11 (1998), 1049-1058.
24. T.F. Xie and S.P. Zhou, Approximation Theory of Real Functions, Hangzhou University Press, Hangzhou, 1998.
25. Z.B. Xu and F.L. Cao, The essential order of approximation for neural networks, Science in China (Ser. F), 47 (2004), 97-112.