Abstract

Neural network models are receiving increasing attention because of their collective computational capabilities. We evaluate the use of the Hopfield neural network model in optically determining the nearest-neighbor of a binary bipolar test vector from a set of binary bipolar reference vectors. The use of the Hopfield model is compared with that of a direct technique called direct storage nearest-neighbor that accomplishes the task of nearest-neighbor determination.

© 1986 Optical Society of America

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References

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  1. J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
    [CrossRef] [PubMed]
  2. D. Psaltis, N. Farhat, “Optical Information Processing Based on an Associative-Memory Model for Neural Nets with Thresholding and Feedback,” Opt. Lett. 10, 98 (1985).
    [CrossRef] [PubMed]
  3. N. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical Implementation of the Hopfield Model,” Appl. Opt. 24, 1469 (1985).
    [CrossRef] [PubMed]
  4. Workshop on Neural Network Models for Computing, Santa Barbara, CA, 30 Apr.–2 May 1985.
  5. Conference on Neural Networks for Computing, Snowbird, UT, 13–16 Apr. 1986.
  6. Y. Abu-Mostafa, J. St. Jacques, “Information Capacity of the Hopfield Model,” IEEE Trans. Inf. Theory IT-31, 461 (1985).
    [CrossRef]
  7. S. S. Venkatesh, D. Psaltis, “Information Storage and Retrieval in Two Associative Nets,” submitted to IEEE Trans. Inf. Theory (1985).
  8. R. J. McEliece, E. C. Posner, E. R. Rodemich, S. S. Venkatesh, “The Capacity of the Hopfield Associative Memory,” submitted to IEEE Trans. Inf. Theory (1986).
  9. R. J. McEliece, E. C. Posner, “The Number of Stable Points of an Infinite-Range Spin Glass Memory,” in Telecommunications and Data Acquisition Progress Report, Vol. 42–83, July–Sept. 1985, (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, 15Nov.1985), pp. 209–215.
  10. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).
  11. R. J. McEliece, The Theory of Information and Coding (Addison-Wesley, Reading, MA, 1977), Chap. 7.
  12. L. O’Gorman, A. C. Sanderson, “The Converging Squares Algorithm: an Efficient Method for Locating Peaks in Multidimensions,” IEEE Tran. Pattern. Anal. Machine. Intell. PAMI-6, 280 (1984).
    [CrossRef]
  13. A. W. Lohmann, W. Stork, G. Stucke, “Optical Perfect Shuffle,” Appl. Opt. 25, 1530 (1986).
    [CrossRef] [PubMed]

1986

1985

1984

L. O’Gorman, A. C. Sanderson, “The Converging Squares Algorithm: an Efficient Method for Locating Peaks in Multidimensions,” IEEE Tran. Pattern. Anal. Machine. Intell. PAMI-6, 280 (1984).
[CrossRef]

1982

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

Abu-Mostafa, Y.

Y. Abu-Mostafa, J. St. Jacques, “Information Capacity of the Hopfield Model,” IEEE Trans. Inf. Theory IT-31, 461 (1985).
[CrossRef]

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Farhat, N.

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Hopfield, J. J.

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

Lohmann, A. W.

McEliece, R. J.

R. J. McEliece, E. C. Posner, “The Number of Stable Points of an Infinite-Range Spin Glass Memory,” in Telecommunications and Data Acquisition Progress Report, Vol. 42–83, July–Sept. 1985, (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, 15Nov.1985), pp. 209–215.

R. J. McEliece, The Theory of Information and Coding (Addison-Wesley, Reading, MA, 1977), Chap. 7.

R. J. McEliece, E. C. Posner, E. R. Rodemich, S. S. Venkatesh, “The Capacity of the Hopfield Associative Memory,” submitted to IEEE Trans. Inf. Theory (1986).

O’Gorman, L.

L. O’Gorman, A. C. Sanderson, “The Converging Squares Algorithm: an Efficient Method for Locating Peaks in Multidimensions,” IEEE Tran. Pattern. Anal. Machine. Intell. PAMI-6, 280 (1984).
[CrossRef]

Paek, E.

Posner, E. C.

R. J. McEliece, E. C. Posner, E. R. Rodemich, S. S. Venkatesh, “The Capacity of the Hopfield Associative Memory,” submitted to IEEE Trans. Inf. Theory (1986).

R. J. McEliece, E. C. Posner, “The Number of Stable Points of an Infinite-Range Spin Glass Memory,” in Telecommunications and Data Acquisition Progress Report, Vol. 42–83, July–Sept. 1985, (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, 15Nov.1985), pp. 209–215.

Prata, A.

Psaltis, D.

Rodemich, E. R.

R. J. McEliece, E. C. Posner, E. R. Rodemich, S. S. Venkatesh, “The Capacity of the Hopfield Associative Memory,” submitted to IEEE Trans. Inf. Theory (1986).

Sanderson, A. C.

L. O’Gorman, A. C. Sanderson, “The Converging Squares Algorithm: an Efficient Method for Locating Peaks in Multidimensions,” IEEE Tran. Pattern. Anal. Machine. Intell. PAMI-6, 280 (1984).
[CrossRef]

St. Jacques, J.

Y. Abu-Mostafa, J. St. Jacques, “Information Capacity of the Hopfield Model,” IEEE Trans. Inf. Theory IT-31, 461 (1985).
[CrossRef]

Stork, W.

Stucke, G.

Venkatesh, S. S.

S. S. Venkatesh, D. Psaltis, “Information Storage and Retrieval in Two Associative Nets,” submitted to IEEE Trans. Inf. Theory (1985).

R. J. McEliece, E. C. Posner, E. R. Rodemich, S. S. Venkatesh, “The Capacity of the Hopfield Associative Memory,” submitted to IEEE Trans. Inf. Theory (1986).

Appl. Opt.

IEEE Tran. Pattern. Anal. Machine. Intell.

L. O’Gorman, A. C. Sanderson, “The Converging Squares Algorithm: an Efficient Method for Locating Peaks in Multidimensions,” IEEE Tran. Pattern. Anal. Machine. Intell. PAMI-6, 280 (1984).
[CrossRef]

IEEE Trans. Inf. Theory

Y. Abu-Mostafa, J. St. Jacques, “Information Capacity of the Hopfield Model,” IEEE Trans. Inf. Theory IT-31, 461 (1985).
[CrossRef]

Opt. Lett.

Proc. Natl. Acad. Sci. USA

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

Other

Workshop on Neural Network Models for Computing, Santa Barbara, CA, 30 Apr.–2 May 1985.

Conference on Neural Networks for Computing, Snowbird, UT, 13–16 Apr. 1986.

S. S. Venkatesh, D. Psaltis, “Information Storage and Retrieval in Two Associative Nets,” submitted to IEEE Trans. Inf. Theory (1985).

R. J. McEliece, E. C. Posner, E. R. Rodemich, S. S. Venkatesh, “The Capacity of the Hopfield Associative Memory,” submitted to IEEE Trans. Inf. Theory (1986).

R. J. McEliece, E. C. Posner, “The Number of Stable Points of an Infinite-Range Spin Glass Memory,” in Telecommunications and Data Acquisition Progress Report, Vol. 42–83, July–Sept. 1985, (Jet Propulsion Laboratory, California Institute of Technology, Pasadena, 15Nov.1985), pp. 209–215.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

R. J. McEliece, The Theory of Information and Coding (Addison-Wesley, Reading, MA, 1977), Chap. 7.

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Figures (3)

Fig. 1
Fig. 1

Sum of outer products memory matrix for the three reference vectors in Eq. (10).

Fig. 2
Fig. 2

Sum of outer products matrix constructed from the three vectors in Eq. (13).

Fig. 3
Fig. 3

Three reference vectors used for the HM in Ref. 3.

Equations (20)

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w i j = w j i for 1 i , j N , w i i = 0 for 1 i N . }
s i = Sgn [ j = 1 N w i j s j - t i ] for 1 i N ,
Sgn [ x ] = { + 1 for x 0 , - 1 for x < 0.
w i j = { m = 1 M u i m u j m 0 for i j for i = j ,             1 i , j N ,
u m = Sgn [ Wu m ] ,
u i m = Sgn [ u i m ( N - 1 ) + k = 1 , k m M j = 1 , j i N u i k u j k u j m ] .
v k + 1 = Wv k , k 1.
s i 1 = Sgn [ j = 1 N w i j s j 1 ] = Sgn [ j = 1 , j i N w i j s j 1 + 0 ] = Sgn [ j = 1 , j i N w i j s j 2 + 0 ] = Sgn [ j = 1 N w i j s j 2 ] = s i 2 .
v j = { - u j if x j 0 , u j = 1 if x j = 0 ,
u i 1 = 1 for 1 i N , u i 2 = { - 1 for 1 i ( N / 2 ) + 1 otherwise , u i 3 = { - 1 for ( 1 + N 4 ) i 3 N 4 + 1 otherwise .
u i 1 = + 1 for 1 i N , u i 2 = { - 1 for 1 i ( N / 2 ) + 1 for ( 1 + N 2 ) i N , u i 3 = { - 1 for 1 i ( α + N 2 ) , + 1 for ( 1 + α + N 2 ) i N ,
v = [ - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 1 1 - 1 1 1 1 1 1 ] T ,
u i = { - 1 for 1 i 16 , i = 22 , 23 , + 1 otherwise .
u = xG ,
T i j = v C v i v j = 2 K .
2 ( K - m ) i = 0 [ m / 2 ] ( m 2 i ) = 2 ( K - 1 )
A T = [ u 1 u 2 u M ] ,
z = Av .
z i = u i T v = a i - d i ,
d ( u i , v ) = d i = ½ ( N - z i ) .

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