# Approximation by superpositions of a sigmoidal function

- 4.2k Downloads
- 4.7k Citations

## Abstract

In this paper we demonstrate that finite linear combinations of compositions of a fixed, univariate function and a set of affine functionals can uniformly approximate any continuous function of*n* real variables with support in the unit hypercube; only mild conditions are imposed on the univariate function. Our results settle an open question about representability in the class of single hidden layer neural networks. In particular, we show that arbitrary decision regions can be arbitrarily well approximated by continuous feedforward neural networks with only a single internal, hidden layer and any continuous sigmoidal nonlinearity. The paper discusses approximation properties of other possible types of nonlinearities that might be implemented by artificial neural networks.

## Key words

Neural networks Approximation Completeness## Preview

Unable to display preview. Download preview PDF.

## References

- [A] R. B. Ash,
*Real Analysis and Probability*, Academic Press, New York, 1972.Google Scholar - [BH] E. Baum and D. Haussler, What size net gives valid generalization?,
*Neural Comput*. (to appear).Google Scholar - [B] B. Bavarian (ed.), Special section on neural networks for systems and control,
*IEEE Control Systems Mag.*,**8**(April 1988), 3–31.Google Scholar - [BEHW] A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth, Classifying learnable geometric concepts with the Vapnik-Chervonenkis dimension,
*Proceedings of the 18th Annual ACM Symposium on Theory of Computing*, Berkeley, CA, 1986, pp. 273–282.Google Scholar - [BST] L. Brown, B. Schreiber, and B. A. Taylor, Spectral synthesis and the Pompeiu problem,
*Ann. Inst. Fourier (Grenoble)*,**23**(1973), 125–154.MathSciNetGoogle Scholar - [CD] S. M. Carroll and B. W. Dickinson, Construction of neural nets using the Radon transform, preprint, 1989.Google Scholar
- [C] G. Cybenko, Continuous Valued Neural Networks with Two Hidden Layers are Sufficient, Technical Report, Department of Computer Science, Tufts University, 1988.Google Scholar
- [DS] P. Diaconis and M. Shahshahani, On nonlinear functions of linear combinations,
*SIAM J. Sci. Statist. Comput.*,**5**(1984), 175–191.CrossRefMathSciNetzbMATHGoogle Scholar - [F] K. Funahashi, On the approximate realization of continuous mappings by neural networks,
*Neural Networks*(to appear).Google Scholar - [G] L. J. Griffiths (ed.), Special section on neural networks,
*IEEE Trans. Acoust. Speech Signal Process.*,**36**(1988), 1107–1190.Google Scholar - [HSW] K. Hornik, M. Stinchcombe, and H. White, Multi-layer feedforward networks are universal approximators, preprint, 1988.Google Scholar
- [HL1] W. Y. Huang and R. P. Lippmann, Comparisons Between Neural Net and Conventional Classifiers, Technical Report, Lincoln Laboratory, MIT, 1987.Google Scholar
- [HL2] W. Y. Huang and R.P. Lippmann, Neural Net and Traditional Classifiers, Technical Report, Lincoln Laboratory, MIT, 1987.Google Scholar
- [J] L. K. Jones, Constructive approximations for neural networks by sigmoidal functions, Technical Report Series, No. 7, Department of Mathematics, University of Lowell, 1988.Google Scholar
- [K] A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition,
*Dokl. Akad. Nauk. SSSR*,**114**(1957), 953–956.MathSciNetzbMATHGoogle Scholar - [LF] A. Lapedes and R. Farber, Nonlinear Signal Processing Using Neural Networks: Prediction and System Modeling, Technical Report, Theoretical Division, Los Alamos National Laboratory, 1987.Google Scholar
- [L1] R. P. Lippmann, An introduction to computing with neural nets,
*IEEE ASSP Mag.*,**4**(April 1987), 4–22.CrossRefGoogle Scholar - [L2] G. G. Lorentz, The 13th problem of Hilbert, in
*Mathematical Developments Arising from Hilbert’s Problems*(F. Browder, ed.), vol. 2, pp. 419–430, American Mathematical Society, Providence, RI, 1976.Google Scholar - [MSJ] J. Makhoul, R. Schwartz, and A. El-Jaroudi, Classification capabilities of two-layer neural nets.
*Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing*, Glasgow, 1989 (to appear).Google Scholar - [MP] M. Minsky and S. Papert,
*Perceptrons*, MIT Press, Cambridge, MA, 1969.zbMATHGoogle Scholar - [N] N. J. Nilsson,
*Learning Machines*, McGraw-Hill, New York, 1965.zbMATHGoogle Scholar - [P] G. Palm, On representation and approximation of nonlinear systems, Part II: Discrete systems,
*Biol. Cybernet.*,**34**(1979), 49–52.CrossRefMathSciNetzbMATHGoogle Scholar - [R1] W. Rudin,
*Real and Complex Analysis*, McGraw-Hill, New York, 1966.zbMATHGoogle Scholar - [R2] W. Rudin,
*Functional Analysis*, McGraw-Hill, New York, 1973.zbMATHGoogle Scholar - [RHM] D. E. Rumelhart, G. E. Hinton, and J. L. McClelland, A general framework for parallel distributed processing, in
*Parallel Distributed Processing: Explorations in the Microstructure of Cognition*(D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, eds.), vol. 1, pp. 45–76, MIT Press, Cambridge, MA, 1986.Google Scholar - [V] L. G. Valiant, A theory of the learnable,
*Comm. ACM*,**27**(1984), 1134–1142.CrossRefzbMATHGoogle Scholar - [WL] A Wieland and R. Leighton, Geometric analysis of neural network capabilities,
*Proceedings of IEEE First International Conference on Neural Networks*, San Diego, CA, pp. III-385–III-392, 1987.Google Scholar