Digital Signal Processing Reference
In-Depth Information
Like the other neural networks, the Hopfield network has the follow-
ing four components:
Neurons:
The Hopfield network has a finite set of neurons
x
(
i
)
,
1
≤
i
N
which serve as processing units. Each neuron has a value (or
state) at time
t
, described by
x
t
(
i
). A neuron in the Hopfield network
has one of the two states, either -1 or +1; that is,
x
t
(
i
)
≤
.
Synaptic connections:
The learned information of a neural net-
work resides within the interconnections between its neurons. For each
pair of neurons
x
(
i
)and
x
(
j
), there is a connection
w
ij
, called the
synapse, between them. The design of the Hopfield network requires that
w
ij
=
w
ji
and
w
ii
= 0. Figure 6.13a illustrates a three-node network.
Propagation rule:
It defines how states and synapses influence the
input of a neuron. The propagation rule
τ
t
(
i
)isdefinedby
∈{−
1
,
+1
}
N
τ
t
(
i
)=
x
t
(
j
)
w
ij
+
b
i
(6.40)
j=1
b
i
is the externally applied bias to the neuron.
Activation function:
The activation function
f
determines the
next state of the neuron
x
t+1
(
i
) based on the value
τ
t
(
i
)computedby
the propagation rule and the current value
x
t
(
i
). Figure 6.13b illustrates
this. The activation function for the Hopfield network, is the hard limiter
defined here:
x
t+1
(
i
)=
f
(
τ
t
(
i
)
,
x
t
(
i
)) =
1
,
if
τ
t
(
i
)
>
0
(6.41)
−
1
,
if
τ
t
(
i
)
<
0
The network learns patterns that are
N
-dimensional vectors from the
space
P
=
N
.Let
e
k
=[
e
k
1
,e
k
2
,...,e
n
]definethe
k
th exemplar
{−
1
,
1
}
pattern where 1
K
. The dimensionality of the pattern space is
reflected in the number of nodes in the network, such that the latter will
have
N
nodes
x
(1)
,
x
(2)
,...,
x
(
N
).
≤
k
≤
The training algorithm of the Hopfield neural network is simple and
outlined below.
1.
Learning:
Assign weights
w
ij
to the synaptic connections:
w
ij
=
k=1
e
i
e
j
,
if
i
=
j
(6.42)
0
,
if
i
=
j
Keep in mind that
w
ij
=
w
ji
, so it is necessary to perform the preceding
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