Kolmogorov's superposition theorem

[sontag@fermat.rutgers.edu]

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   KOLMOGOROV'S SUPERPOSITION THEOREM AND ARTIFICIAL NEURAL NETWORKS

                    Rui J. P. de Figueiredo
           Dept. of Electrical and Computer Engineering
              Rice University, Houston, TX 77251-1892
                   e-mail: rui at zeta.rice.edu
                

The implementation of the Kolmogorov-Arnold-Sprecher Superposition Theorem 
[1-3] in terms of artificial neural networks was first presented and fully 
discussed by me in 1980 [4]. I also discussed, then [4], applications of
these structures to statistical pattern recognition and image and multi-
dimensional signal processing. However, I did not use the words "neural 
networks" in defining the underlying networks. For this reason, the current
researchers on neural nets including Robert Hecht-Nielsen [5] do not seem to
be aware of my contribution [4]. I hope that this note will help correct
history.

  Incidentally, there is a misprint in [4]. In [4], please insert "no" in
the statement before eqn.(4). That statement should read: "Sprecher showed
that lambda can be any nonzero number which satisfies no equation ..."

[1] A.K.Kolmogorov, "On the representation of continuous functions of several
    variables by superposition of continuous functions of one variable and 
    addition," Dokl.Akad.Nauk.SSSR,Vol.114,pp.369-373,1957.
[2] V.I.Arnol'd, "On functions of three variables," Dokl.Akad.Nauk.SSSR, 
    Vol.114,pp.953-956,1957.
[3] D.A.Sprecher, "An improvement in the superposition theorem of Kolmogorov,"
    J.Math.Anal.Appl.,Vol.38,pp.208-213,1972.
[4] Rui J.P.de Figueiredo, "Implications and applications of Kolmogorov's
    superposition theorem,"IEEE Trans.Auto.Contr.,Vol.AC-25,pp.1227-1231,1980.
[5] R.Hecht-Nielsen, "Kolmogorov's mapping neural network existence theorem,"
    IEEE 1st Int.Conf.on Neural Networks, San Diego,CA,June 21-24,1987,paper
    III-11.