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A Constructive Proof and An Extension of Cybenko’s Approximation Theorem

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Abstract

In this paper, we present a constructive proof of approximation by superposition of sigmoidal functions. We point out a sufficient condition that the set of finite linear combinations of the form \(\sum \alpha _j\sigma (y_jx+\theta _j)\) is dense in \(C(\mathbb{I}^n)\), is the boundedness of the sigmoidal function σ(x). Moreover, we show that if the set of finite linear combinations of the form \(\sum c_j\omega (\xi _j+\eta _j)\), where ω is a univariate function, is dense in \(L^p[a,b] (1\leq p< \infty )\) (or C[a,b]) for any finite a,b, then the set of finite linear combinations of the form \(\sum c_j\omega (y_j.x+\theta _j)\) is dense in \(L^p(\mathbb{I}^n)(or C(\mathbb{I}^n))\). An extension in another direction is also presented in Theorem 4 of this paper.

Key words

  • Constructive approximation
  • Neural networks
  • Sigmoidal functions

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  • DOI: 10.1007/978-1-4612-2856-1_21
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References

  1. G. Cybenko, “Approximation by Superpositions of a Sigmoidal Function”, Mathematics of Control, Signals and Systems, V.2, No.4 (1989), P.303–314.

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  2. A.N. Komogorov, “On the Representation of Continuous Functions of Several Variables by Superposition of Continuous Functions of One Variable and Addition”, Dokl. Akad. Nawk. SSSR, 114, 1957 (pp.953–956) English translation, American Math. Soc. Transl (2), 28 (1963), pp.55–59, MR.22#2669; 27#3760.

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  3. E.M. Stein and Guido Weiss, “Introduction to Fourier Analysis on Euclidean Spaces”, Princeton University Press (1971).

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  4. L.K. Jones, “Constructive Approximation for Neural Networks by Sigmoidal Functions”, Technical Report Series, No.7, Dept. of Mathematics, University of Lowell (1988).

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  5. S.M. Carrol and B.W. Dickinson, “Construction of Neural Nets Using Radon Transform”, Preprint 1989.

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  6. K. Funahashi, “On the Approximate Realization of Continuous Mappings by Neural Networks”, Journal of International Neural Networks (to appear).

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  7. K. Hornik, M. Stinchcombe and H. White, “Multilayer Feedforward Networks are Universal Approximators”, Preprint 1988.

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© 1992 Springer-Verlag New York, Inc.

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Chen, T., Chen, H., Liu, Rw. (1992). A Constructive Proof and An Extension of Cybenko’s Approximation Theorem. In: Page, C., LePage, R. (eds) Computing Science and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2856-1_21

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  • DOI: https://doi.org/10.1007/978-1-4612-2856-1_21

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-97719-5

  • Online ISBN: 978-1-4612-2856-1

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