Abstract
For many problems in classification, compensation, adaptivity, identification, and signal processing, results concerning the representation and approximation of nonlinear functions can be of particular interest to engineers. Here we consider a large class of functionsf that map ℝn into the set of real or complex numbers, and we give bounds on the number of parameters of a certain approximation network so thatf can be approximated to within a prescribed degree of accuracy using an appropriate configuration of the network. We also describe related work in the neural networks literature.
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References
A. R. Barron, Universal approximation bounds for superpositions of a sigmoidal function,IEEE Transactions on Information Theory, vol. 39, no. 3, pp. 930–945, May 1993.
S. Bochner and K. Chandrasekharan,Fourier Transforms, Princeton University Press, Princeton, NJ, 1949.
G. Cybenko, Approximation by superposition of a single function,Mathematics of Control, Signals and Systems, vol. 2, pp. 303–314, 1989.
A. T. Dingankar and I. W. Sandberg, A note on error bounds for approximation in inner product spaces,Circuits, Systems, and Signal Processing, vol. 15, no. 4, pp. 519–522, 1996.
D. Docampo, C. T. Abdallah, and D. Hush, Error bounds in constructive approximation,Proceedings of the Fourth Bayona Workshop on Intelligent Methods in Signal Processing and Communications, Bayona, Spain, June 1996, pp. 24–28.
K. Funahashi, On the approximate realization of continuous mappings by neural networks,IEEE Transactions on Neural Networks, vol. 2, pp. 183–192, 1989.
F. Girosi and G. Anzellotti, Rates of convergence for radial basis functions and neural networks,Artificial Neural Networks with Applications in Speech and Vision, Chapman & Hall, New York 1993.
L. K. Jones, A simple lemma on greedy approximation in Hilbert space and convergence, rates for projection pursuit regression and neural network training,Annals of Statistics, vol. 20, no. 1, pp. 608–613, March 1992.
H. N. Mhaskar and C. A. Micchelli, Approximation by superposition of sigmoidal and radial basis functions,Advances in Applied Mathematics, vol. 3, pp. 350–373, 1992.
G. Pisier, Remarques sur un resultat non publié de B. Maurey,Sem. d'Analyse Fonctionelle, vol. 1, no. 12, Ecole Poly. Centre de Math., Palaiseau, France, 1980–1981.
H. L. Royden,Real Analysis, Macmillan, New York, 1968.
I. W. Sandberg, Approximations for nonlinear functionals,IEEE Transactions on Circuits and Systems-I. Fundamental Theory and Applications, vol. 39, no. 1, pp. 65–67, January 1992.
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Dingankar, A.T., Sandberg, I.W. A note on error bounds for function approximation using nonlinear networks. Circuits Systems and Signal Process 17, 449–457 (1998). https://doi.org/10.1007/BF01201501
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DOI: https://doi.org/10.1007/BF01201501