Abstract

We consider the convergence characteristics of a perceptron learning algorithm, taking into account the decay of photorefractive holograms during the process of interconnection weight changes. As a result of the hologram erasure, the convergence of the learning process is dependent on the exposure time during the weight changes. A mathematical proof of the conditional convergence, as well as computer simulations of the photorefractive perceptrons, is presented and discussed.

© 1993 Optical Society of America

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References

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  1. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 5, p. 142.
  2. D. Psaltis, D. Brady, K. Wagner, Appl. Opt. 27, 1752 (1988).
    [CrossRef]
  3. J. Hong, S. Campbell, P. Yeh, Appl. Opt. 29, 3019 (1990).
    [CrossRef] [PubMed]
  4. K. Y. Hsu, S. H. Lin, C. J. Cheng, T. C. Hsieh, P. Yeh, Int. J. Opt. Comput. (to be published).
  5. E. G. Paek, J. R. Wullert, J. S. Patel, Opt. Lett. 14, 1303 (1989).
    [CrossRef] [PubMed]
  6. See, for example,P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), App. A, pp. 392–393.

1990 (1)

1989 (1)

1988 (1)

Brady, D.

Campbell, S.

Cheng, C. J.

K. Y. Hsu, S. H. Lin, C. J. Cheng, T. C. Hsieh, P. Yeh, Int. J. Opt. Comput. (to be published).

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 5, p. 142.

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 5, p. 142.

Hong, J.

Hsieh, T. C.

K. Y. Hsu, S. H. Lin, C. J. Cheng, T. C. Hsieh, P. Yeh, Int. J. Opt. Comput. (to be published).

Hsu, K. Y.

K. Y. Hsu, S. H. Lin, C. J. Cheng, T. C. Hsieh, P. Yeh, Int. J. Opt. Comput. (to be published).

Lin, S. H.

K. Y. Hsu, S. H. Lin, C. J. Cheng, T. C. Hsieh, P. Yeh, Int. J. Opt. Comput. (to be published).

Paek, E. G.

Patel, J. S.

Psaltis, D.

Wagner, K.

Wullert, J. R.

Yeh, P.

J. Hong, S. Campbell, P. Yeh, Appl. Opt. 29, 3019 (1990).
[CrossRef] [PubMed]

K. Y. Hsu, S. H. Lin, C. J. Cheng, T. C. Hsieh, P. Yeh, Int. J. Opt. Comput. (to be published).

See, for example,P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), App. A, pp. 392–393.

Appl. Opt. (2)

Opt. Lett. (1)

Other (3)

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 5, p. 142.

K. Y. Hsu, S. H. Lin, C. J. Cheng, T. C. Hsieh, P. Yeh, Int. J. Opt. Comput. (to be published).

See, for example,P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), App. A, pp. 392–393.

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Figures (3)

Fig. 1
Fig. 1

Graphic method for finding solutions of inequality (10). Intersection occurs only when the condition of inequality (13) is satisfied.

Fig. 2
Fig. 2

Pictures of our training patterns. The sampling grid of the computer simulation is 32 × 32.

Fig. 3
Fig. 3

Number of steps (open circles, number of weight changes; filled circles, number of training cycles) for photorefractive perceptron convergence as functions of the normalized exposure time t/τ.

Tables (1)

Tables Icon

Table 1 Computer Simulation of Photorefractive Perceptrons with Different t/τ Values

Equations (15)

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w ( k + 1 ) = w ( k ) exp [ | σ ( k ) | t / τ ] + σ ( k ) [ 1 exp ( t / τ ) ] x ( k ) ,
σ ( k ) = { 0 if x ( k ) is correctly classified 1 if x ( k ) C 1 but w ( k ) x ( k ) < θ , 1 if x ( k ) C 2 but w ( k ) x ( k ) > θ }
y = { x x C 1 x x C 2
w ( p + 1 ) = w ( p ) exp ( t / τ ) + [ 1 exp ( t / τ ) ] y ( p ) .
w ( p + 1 ) = [ 1 exp ( t / τ ) ] k = 1 p y ( k ) exp [ ( p k ) t / τ ] .
| w ( p + 1 ) | 2 > [ 1 exp ( p t / τ ) ] 2 β 2 | w | 2 ,
β = min 1 k M w y ( k ) .
| w ( p + 1 ) | 2 < [ 1 exp ( t / τ ) ] 2 α [ 1 exp ( 2 p t / τ ) ] [ 1 exp ( 2 t / τ ) ] ,
α = max 1 k M | y ( k ) | 2 .
tanh ( p t / 2 τ ) tanh ( t / 2 τ ) < α | w | 2 β 2 P i max ,
tanh ( p 0 t / 2 τ ) tanh ( t / 2 τ ) < α | w | 2 β 2 P i max ,
p < P i max .
tanh ( t / 2 τ ) P i max < 1 .
( t / 2 τ ) P i max < 1 .
( t / 2 τ ) P i max > 1 ,

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