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0066_Frame_C32.fm  Page 18  Wednesday, January 9, 2002  7:54 PM










                                                  v 1                      v 1
                                                           W

                                                v 2                        v 2

                                                 v 3                       v 3


                                                 v 4                       v 4


                                                 v 5                      v 5





                       FIGURE 32.17  A Hopfield network or autoassociative memory.

                       programmed by adjusting the weights using a modified Hebbian rule,
                                                 ∆w ij =  ∆w ji =  ( 2v i –  1) 2v j –  1)c     (32.42)
                                                                    (

                       Such memory has limited storage capacity. Based on experiments, Hopfield estimated that the maximum
                       number of stored patterns is 0.15N, where N is the number of neurons.
                         Later the concept of energy function was extended by Hopfield (1984) to one-layer recurrent networks
                       having neurons with continuous activation functions. These types of networks were used to solve many
                       optimization and linear programming problems.

                       Autoassociative Memory
                       Hopfield (1984) extended the concept of his network to autoassociative memories. In the same network
                       structure as shown in Fig. 32.17, the bipolar hard-threshold neurons were used with outputs equal to −1
                       or +1. In this network, pattern s m  are stored into the weight matrix W using the autocorrelation algorithm

                                                            M
                                                      W =  ∑  s m s m –  MI                     (32.43)
                                                                 T
                                                           m=1
                       where M is the number of stored patterns and I is the unity matrix. Note that W is the square symmetrical
                       matrix with elements on the main diagonal equal to zero (w ji  for i = j). Using a modified formula (32.42),
                       new patterns can be added or subtracted from memory. When such memory is exposed to a binary
                       bipolar pattern by enforcing the initial network states, after signal circulation the network will converge
                       to the closest (most similar) stored pattern or to its complement. This stable point will be at the closest
                       minimum of the energy

                                                       E v() =  –  1 T                          (32.44)
                                                               --v Wv
                                                               2
                       Like the Hopfield network, the autoassociative memory has limited storage capacity, which is estimated
                       to be about M max  = 0.15N. When the number of stored patterns is large and close to the memory capacity,
                       the network has a tendency to converge to spurious states, which were not stored. These spurious states
                       are additional minima of the energy function.




                       ©2002 CRC Press LLC
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