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THE HOPFIELD NETWORK
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fundamental memory ð 1; 1; 1Þ attracts unstable states ð 1; 1; 1Þ, ð 1; 1; 1Þ
and ð1; 1; 1Þ. Here again, each of the unstable states represents a single error,
compared to the fundamental memory. Thus, the Hopfield network can indeed
act as an error correction network. Let us now summarise the Hopfield network
training algorithm.
Step 1:
Storage
The n-neuron Hopfield network is required to store a set of M fundamental
memories, Y 1 ; Y 2 ; ...; Y M . The synaptic weight from neuron i to
neuron j is calculated as
8
><
X M
y m;i y m;j ; i 6¼ j
w ij ¼
; ð6:22Þ
m¼1 >:
0; i ¼ j
where y m;i and y m;j are the ith and the jth elements of the fundamental
memory Y m , respectively. In matrix form, the synaptic weights
between neurons are represented as
W ¼ XM
m¼1
Y m Y T m
MI
The Hopfield network can store a set of fundamental memories if the
weight matrix is symmetrical, with zeros in its main diagonal (Cohen
and Grossberg, 1983). That is,
2
3
0 w 12 w 1i w 1n
w 21 0 w 2i w 2n
.
.
. . . W ¼
w i1 w i2 0 w ; ð6:23Þ
in
6 . . . .
7
4 . . . . 5
w n1 w n2 w ni 0
where w ij ¼ w ji .
Once the weights are calculated, they remain fixed.
Step 2:
Testing
We need to confirm that the Hopfield network is capable of recalling all
fundamental memories. In other words, the network must recall any
fundamental memory Y m when presented with it as an input. That is,
x m;i ¼ y m;i ; i ¼ 1; 2; ...; n; m ¼ 1; 2; ...; M
0
1
y m;i ¼ sign@
Xn
j¼1
w ij x m;j
i
A;