Information Technology Reference
In-Depth Information
It has been shown that update of the weights based on the
increase
and
decrease
operators satisfies two basic postulates of quantum mechanics, i.e.: (a) existence of
the stochastic weights
w
ij
in a superposition of states, (b) evolution of the weights
with the use of unitary operators, i.e. R
i
n
ıR
n
D R
n
ıR
i
n
D I [
149
,
165
]. Moreover,
it has been shown that the aforementioned operators are equivalent to quantum
addition and quantum subtraction, respectively [
163
,
165
].
10.2
Attractors in QHO-Based Associative Memories
10.2.1
Decomposition of the Weight Matrix
into a Superposition of Matrices
Taking the weights
w
ij
of the weight matrix W to be stochastic variables with
p.d.f. (or possibility distribution) as the one depicted in Fig.
7.1
b means that W
can be decomposed into a superposition of associative memories (Fig.
10.3
). The
equivalence of using probabilistic or possibilistic (fuzzy) variables in the description
of uncertainty has been analyzed in [
51
].
Following the stochastic (fuzzy) representation of the neural weights, the overall
associative memory W equals a weighted averaging of the individual weight
matrices W
i
, i.e. W D
P
iD1
i
W
i
, where the nonnegative weights
i
indicate the
contribution of each local associative memory W
i
to the aggregate outcome [
165
].
Without loss of generality, a 3 3 weight matrix of a neural associative memory
is considered. It is also assumed that the weights
w
ij
are stochastic (fuzzy) variables
and that quantum learning has been used for the weights update. The weights satisfy
the condition
P
iD1
A
i
.
w
ij
/ D 1 (strong fuzzy partition). Thus, the following
combinations of membership values of the elements of the matrices
W
i
are possible:
W
1
W
12
13
23
W
5
W 1
12
13
23
W
2
W
12
13
1
23
W
6
W 1
12
13
1
23
W
3
W
12
1
13
23
W
7
W 1
12
1
13
23
W
4
W
12
1
13
1
23
W
8
W 1
12
1
13
1
23
W
i
gives
The decomposition of matrix W into a set of superimposing matrices
8
<
9
=
0
1
0
1
0a
A
i
12
a
A
i
l
12
13
12
23
13
23
13
W
1
D
@
A
;
@
a
A
i
12
0a
A
i
A
:
23
;
a
A
i
13
a
A
i
23
0
8
<
0
@
1
A
9
=
0
1
0
A
i
12
a
A
i
13
l
12
13
W
2
D
@
A
;
a
A
i
12
0
A
i
C
1
23
12
1
23
:
;
13
a
A
i
C
1
a
A
i
13
1
23
0
23
