Abstract

By combining Kohonen learning and Grossberg learning a new type of mapping neural network is obtained. This counterpropagation network (CPN) functions as a statistically optimal self-programming lookup table. The paper begins with some introductory comments, followed by the definition of the CPN. Then a closed-form formula for the error of the network is developed. The paper concludes with a discussion of CPN variants and comments about CPN convergence and performance. References and a neurocomputing bibliography with a combined total of eighty entries are provided.

© 1987 Optical Society of America

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References

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  1. D. E. Rumelhart, J. L. McClelland, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vols. 1, 2, 3 (MIT Press, Cambridge, MA, 1986, 1987).
  2. T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, New York, 1984).
  3. S. Grossberg, Studies of Mind and Brain (Reidel, Boston, MA, 1982).
    [CrossRef]
  4. R. Hecht-Nielsen, “Combinatorial Hypercompression,” in Proceedings, IEEE International Conference on Neural Networks (IEEE, New York, 1987).

Grossberg, S.

S. Grossberg, Studies of Mind and Brain (Reidel, Boston, MA, 1982).
[CrossRef]

Hecht-Nielsen, R.

R. Hecht-Nielsen, “Combinatorial Hypercompression,” in Proceedings, IEEE International Conference on Neural Networks (IEEE, New York, 1987).

Kohonen, T.

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, New York, 1984).

McClelland, J. L.

D. E. Rumelhart, J. L. McClelland, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vols. 1, 2, 3 (MIT Press, Cambridge, MA, 1986, 1987).

Rumelhart, D. E.

D. E. Rumelhart, J. L. McClelland, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vols. 1, 2, 3 (MIT Press, Cambridge, MA, 1986, 1987).

Other (4)

D. E. Rumelhart, J. L. McClelland, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vols. 1, 2, 3 (MIT Press, Cambridge, MA, 1986, 1987).

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, New York, 1984).

S. Grossberg, Studies of Mind and Brain (Reidel, Boston, MA, 1982).
[CrossRef]

R. Hecht-Nielsen, “Combinatorial Hypercompression,” in Proceedings, IEEE International Conference on Neural Networks (IEEE, New York, 1987).

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Figures (5)

Fig. 1
Fig. 1

Topology of the counterpropagation network.

Fig. 2
Fig. 2

CPN mathematics.

Fig. 3
Fig. 3

Kohonen's weight change law.

Fig. 4
Fig. 4

The CPN module.

Fig. 5
Fig. 5

Forward-only CPN.

Equations (14)

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F ( W ) = Ω n θ ( W , z ) ρ ( z ) d A ( z ) ,
W = [ w 1 , w 2 , , w N ] z , w i Ω n = { x R n | | x | = 1 } ,
θ ( W , z ) min ( | z w 1 | 2 , | z w 2 | 2 , , | z w N | 2 ) .
| z w i | 2 = | z | 2 + | w i | 2 2 z w i ,
θ ( W , z ) = 2 2 max ( z w 1 , z w 2 , , z w N ) .
B i { z Ω n | i is the smallest integer 1 i N such that z w i z w j j } ,
F ( W ) = 2 2 i = 1 N w i B i z ρ ( z ) d A ( z ) ,
z w i = max ( z w 1 , z w 2 , , z w N ) ,
m i B i ρ ( z ) d A ( z )
v i ( 1 / m i ) B i z ρ ( z ) d A ( z )
F ( W ) = 2 2 i = 1 N m i ( w i v i ) .
i = 1 N m i = 1 .
i = 1 N v i = Ω n z ρ ( z ) d A ( z ) .
α x + ( 1 α ) 1 ] ̂

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