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From: bmoghad@gmuvax2.gmu.edu (bmoghad)
Newsgroups: comp.ai.neural-nets
Subject: Re: Searching for 13th Hilbert Problem article
Keywords: Kolmogorov's Theorem
Message-ID: <2850@gmuvax2.gmu.edu>
Date: 8 Nov 90 23:27:56 GMT
References: <39365@ut-emx.uucp>
Reply-To: bmoghad@gmuvax2.UUCP (Baback Moghaddam)
Distribution: usa
Organization: C3I Center, George Mason University
Lines: 44

In article <39365@ut-emx.uucp> phbd641@ccwf.cc.utexas.edu (David Chao) writes:
>
>In the article on Hilbert's 13th problem, reference is made to 
>Kolmogorov's proof concerning the approximation of any continous function of
>N variables using only linear summations and nonlinear but
>continously increasing functions of only one variable. (shades of
>typical backprop n.n.'s with logistic activation functions!).
>
>If anyone has any information - in particular
>about Kolmogorov's theorem - I'd appreciate an email response.


 I believe you are referring to Kolmogorov's neural network existence
theorem which claims that any continuous non-linear mapping can be achieved
with a 3-layer network. This is the modern-day neural network extension of
Hilbert's 13th problem and other conjectures in functional decomposition
that have been around for a long time.
 
 Here are a few references which should help:

1. Robert Hecht-Nielsen, "Kolmogorov's Mapping Neural Network Existence
   Theorem", Proceedings of the 1st Int'l Conference on Neural Networks",
   San Diego, June 21-24, IEEE, 1987.

2. Kolmogorov's Theorem (1957)
   Transactions of A.M.S., 28, pp. 55-59 (1963)

3. Sprecher's Theorem (1965)
   Transactions of A.M.S., 115, pp. 340-355 

4. Irie-Miyake's Theorem (1988)
   IEEE Int'l J. Conf. on Neural Networks, I, pp.641-648

5. Funahashi's Theorem (1989)
   Neural Networks, 2, pp. 183-192

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Baback Moghaddam                      |
Center for Exellence in C3I           |   bmoghad@gmuvax2.gmu.edu
Dept. of Electrical & Computer Engr.  |
George Mason University               |   Tel: (703) 764-6283
Fairfax, Virginia  22030-4444         |
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