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In-Depth Information
Fig. 10.1
A connectionist associative memory
memory. Moreover, it will be shown that the considered associative memory with
stochastic weights exhibits an exponential increase in the number of attractors,
comparing to conventional associative memories [
83
,
91
,
156
].
Suppose that a set of pN-dimensional vectors (
fundamental memories
)areto
be stored in a Hopfield associative memory of N neurons, as the one depicted in
Fig.
10.1
. The output of neuron j is given by s
j
D
sgn
.v
j
/; j D 1;2; ;N and
the input v
j
is given by v
j
D
P
iD1
w
ji
s
i
j
j D 1;2; ;N, where s
i
are the
outputs of the neurons connected to the j-th neuron,
j
is the threshold, and
w
ji
's
are the associated weights [
78
,
82
].
The weights of the associative memory are selected using Hebb's postulate
of learning, i.e.
w
ji
N
P
p
kD1
x
j
k
x
k
;j¤ i and
w
ji
D 0;j D i, where
i;j D 1;2; ;N. The above can be summarized in the network's correlation
weight matrix which is given by
1
D
p
X
1
N
x
k
T
x
k
W D
(10.1)
kD1
From Eq. (
10.1
) it can be easily seen that in the case of binary fundamental
memories the weights learning is given by Hebb's rule
w
ji
.n C 1/ D
w
ji
.n/ C
sgn
.x
kC1
x
kC1
/
(10.2)
