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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Neural networks for localized approximation


Authors: C. K. Chui, Xin Li and H. N. Mhaskar
Journal: Math. Comp. 63 (1994), 607-623
MSC: Primary 65D15; Secondary 41A15, 41A30, 92B20
DOI: https://doi.org/10.1090/S0025-5718-1994-1240656-2
MathSciNet review: 1240656
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Abstract: We prove that feedforward artificial neural networks with a single hidden layer and an ideal sigmoidal response function cannot provide localized approximation in a Euclidean space of dimension higher than one. We also show that networks with two hidden layers can be designed to provide localized approximation. Since wavelet bases are most effective for local approximation, we give a discussion of the implementation of spline wavelets using multilayered networks where the response function is a sigmoidal function of order at least two.


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  • Andrew R. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inform. Theory 39 (1993), no. 3, 930–945. MR 1237720, DOI https://doi.org/10.1109/18.256500
  • E. K. Blum and L. K. Li, Approximation theory and neural networks, Neural Networks 4 (1991), 511-515. S. M. Caroll and S. M. Dickinson, Construction of neural nets using the Radon transform, preprint, 1990. T. P. Chen, H. Chen, and R. W. Liu, A constructive proof of approximation by superposition of sigmoidal functions for neutral networks, preprint, 1990.
  • Charles K. Chui and Xin Li, Approximation by ridge functions and neural networks with one hidden layer, J. Approx. Theory 70 (1992), no. 2, 131–141. MR 1172015, DOI https://doi.org/10.1016/0021-9045%2892%2990081-X
  • Charles K. Chui and Xin Li, Realization of neural networks with one hidden layer, Multivariate approximation: from CAGD to wavelets (Santiago, 1992) Ser. Approx. Decompos., vol. 3, World Sci. Publ., River Edge, NJ, 1993, pp. 77–89. MR 1359545
  • Charles K. Chui and Jian-zhong Wang, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992), no. 2, 903–915. MR 1076613, DOI https://doi.org/10.1090/S0002-9947-1992-1076613-3
  • G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems 2 (1989), no. 4, 303–314. MR 1015670, DOI https://doi.org/10.1007/BF02551274
  • W. Dahmen and C. A. Micchelli, Some remarks on ridge functions, Approx. Theory Appl. 3 (1987), 139-143. K. I. Funahashi, On the approximate realization of continuous mappings by neural networks, Neural Networks 2 (1989), 183-192. K. Hornik, M. Stinchcombe, and H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (1989), 359-366. B. Irie and S. Miyake, Capabilities of three layered perceptrons, IEEE Internat. Conf. on Neural Networks 1 (1988), 641-648. Y. Ito, Representation of functions by superpositions of a step or sigmoid function and their applications to neural network theory, Neural Networks 4 (1991), 385-394. ---, Approximation of functions on a compact set by finite sums of a sigmoid function without scaling, Neural Networks 4 (1991), 817-826.
  • H. N. Mhaskar, Approximation properties of a multilayered feedforward artificial neural network, Adv. Comput. Math. 1 (1993), no. 1, 61–80. MR 1230251, DOI https://doi.org/10.1007/BF02070821
  • H. N. Mhaskar and Charles A. Micchelli, Approximation by superposition of sigmoidal and radial basis functions, Adv. in Appl. Math. 13 (1992), no. 3, 350–373. MR 1176581, DOI https://doi.org/10.1016/0196-8858%2892%2990016-P
  • T. Poggio and F. Girosi, Regularization algorithms for learning that are equivalent to multilayer networks, Science 247 (1990), no. 4945, 978–982. MR 1038271, DOI https://doi.org/10.1126/science.247.4945.978
  • Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
  • I. J. Schoenberg, Cardinal spline interpolation, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12. MR 0420078
  • M. Stinchcombe and H. White, Universal approximation using feedforward network with non-sigmoid hidden layer activation functions, Proc. Internat. Joint Conference on Neural Networks (1989), 613-618, San Diego, SOS printing. ---, Approximating and learning unknown mappings using multilayer feedforward networks with bounded weights, IEEE Internat. Conf. on Neural Networks 3 (1990), III-7-III-16.

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Additional Information

Keywords: Neural networks, sigmoidal functions, spline functions, wavelets
Article copyright: © Copyright 1994 American Mathematical Society