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.

**[1]**A. R. Barron,*Universal approximation bounds for superposition of a sigmoidal function*, preprint, November 1990. MR**1237720 (94h:92001)****[2]**E. K. Blum and L. K. Li,*Approximation theory and neural networks*, Neural Networks**4**(1991), 511-515.**[3]**S. M. Caroll and S. M. Dickinson,*Construction of neural nets using the Radon transform*, preprint, 1990.**[4]**T. P. Chen, H. Chen, and R. W. Liu,*A constructive proof of approximation by superposition of sigmoidal functions for neutral networks*, preprint, 1990.**[5]**C. K. Chui and X. Li,*Approximation by ridge functions and neural networks with one hidden layer*, J. Approx. Theory**70**(1992), 131-141. MR**1172015 (93d:41018)****[6]**-,*Realization of neural networks with one hidden layer*, Multivariate approximations: From CAGD to Wavelets (K. Jetter and F. Utreras, eds.), World Scientific Publ., Singapore, 1993, pp. 77-89. MR**1359545****[7]**C. K. Chui and J. Z. Wang,*On compactly supported spline wavelets and a duality principle*, Trans. Amer. Math. Soc.**330**(1992), 903-916. MR**1076613 (92f:41020)****[8]**G. Cybenko,*Approximation by superposition of sigmoidal functions*, Math. Control Signals Systems**2**(4) (1989), 303-314. MR**1015670 (90m:41033)****[9]**W. Dahmen and C. A. Micchelli,*Some remarks on ridge functions*, Approx. Theory Appl.**3**(1987), 139-143.**[10]**K. I. Funahashi,*On the approximate realization of continuous mappings by neural networks*, Neural Networks**2**(1989), 183-192.**[11]**K. Hornik, M. Stinchcombe, and H. White,*Multilayer feedforward networks are universal approximators*, Neural Networks**2**(1989), 359-366.**[12]**B. Irie and S. Miyake,*Capabilities of three layered perceptrons*, IEEE Internat. Conf. on Neural Networks**1**(1988), 641-648.**[13]**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.**[14]**-,*Approximation of functions on a compact set by finite sums of a sigmoid function without scaling*, Neural Networks**4**(1991), 817-826.**[15]**H. N. Mhaskar,*Approximation properties of a multilayered feedforward artificial neural network*, Adv. in Comput. Math.**1**(1993), 61-80. MR**1230251 (94h:41020)****[16]**H. N. Mhaskar and C. A. Micchelli,*Approximation by superposition of a signmoidal function*, Adv. in Appl. Math.**13**(1992), 350-373. MR**1176581 (93f:41030)****[17]**T. Poggio and F. Girosi,*Regularization algorithms for learning that are equivalent to multilayer networks*, Science**247**(1990), 978-982. MR**1038271 (90k:92076)****[18]**W. Rudin,*Functional analysis*, McGraw-Hill, New York, 1973. MR**0365062 (51:1315)****[19]**I. J. Schoenberg,*Cardinal spline interpolation*, CBMS-NSF Conf. Series in Appl. Math. #12, SIAM, Philadelphia, PA, 1973. MR**0420078 (54:8095)****[20]**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.**[21]**-,*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

DOI:
https://doi.org/10.1090/S0025-5718-1994-1240656-2

Keywords:
Neural networks,
sigmoidal functions,
spline functions,
wavelets

Article copyright:
© Copyright 1994
American Mathematical Society