AI - a Guide to Intelligent Systems.pdf - Member of EEPIS

194
ARTIFICIAL NEURAL NETWORKS
where y m;i is the ith element of the actual output vector Y m , and x m;j is
the jth element of the input vector X m . In matrix form,
X m ¼ Y m ;
Y m ¼ sign ðWX m
m ¼ 1; 2; ...; M
hÞ
If all fundamental memories are recalled perfectly we may proceed to
the next step.
Step 3:
Retrieval
Present an unknown n-dimensional vector (probe), X, to the network
and retrieve a stable state. Typically, the probe represents a corrupted or
incomplete version of the fundamental memory, that is,
X 6¼ Y m ;
m ¼ 1; 2; ...; M
(a)
Initialise the retrieval algorithm of the Hopfield network by setting
x j ð0Þ ¼x j
j ¼ 1; 2; ...; n
and calculate the initial state for each neuron
0
1
y i ð0Þ ¼sign@
Xn
w ij x j ð0Þ A; i ¼ 1; 2; ...; n
j¼1
i
where x j ð0Þ is the jth element of the probe vector X at iteration
p ¼ 0, and y i ð0Þ is the state of neuron i at iteration p ¼ 0.
In matrix form, the state vector at iteration p ¼ 0 is presented as
Yð0Þ ¼sign ½WXð0Þ
h Š
(b) Update the elements of the state vector, YðpÞ, according to the
following rule:
0
1
y i ðp þ 1Þ ¼sign@
Xn
w ij x j ðpÞ A
j¼1
i
Neurons for updating are selected asynchronously, that is,
randomly and one at a time.
Repeat the iteration until the state vector becomes unchanged,
or in other words, a stable state is achieved. The condition for
stability can be defined as:
0
1
y i ðp þ 1Þ ¼sign@
Xn
w ij y j ðpÞ A; i ¼ 1; 2; ...; n ð6:24Þ
or, in matrix form,
j¼1
i
Yðp þ 1Þ ¼sign ½WYðpÞ h Š ð6:25Þ