Information Technology Reference
In-Depth Information
Fig. 10.2
Partition of the real
axis x 2 R into an infinite
number of symmetric
triangular possibility
distributions (fuzzy sets)
for high dimensional spaces
N
! 0 and the vectors x
i
and x
k
will be practically
orthogonal.
Thus, taking into account the orthogonality of the fundamental memories x
k
and
Lemma
1
, it can be concluded that memory patterns in high dimensional spaces
practically coincide with the eigenvectors of the weight matrix W .
10.1.3
Learning Through Unitary Quantum Mechanical
Operators
As already mentioned, the update of the weights
w
ij
of the associative memories is
performed according to Eq. (
10.2
), which implies that the value of
w
ij
is increased
by or decreased as indicated by
sgn
.x
i
x
j
/. If the weight
w
ij
isassumedtobea
stochastic variable, which is described by the probability (possibility) distribution
of Fig.
7.1
b, then the term
sgn
.x
k
x
k
/ of Hebbian learning can be replaced by a
stochastic increment.
A way to substantiate this stochastic increment is to describe variable
w
ij
in terms of a possibility distribution (fuzzy sets). To this end, the real axis x,
where the
w
ij
takes its values, is partitioned in triangular possibility distributions
(fuzzy sets) A
1
;A
2
; ;A
n1
;A
n
(see Fig.
10.2
). These approximate sufficiently
the Gaussians depicted in Fig.
7.1
b. Then, the
increase
of the fuzzy (stochastic)
weight is performed through the possibility transition matrix R
i
n
which results into
A
n
D R
i
n
ıA
n1
, with ı being the max-min operator. Similarly, the
decrease
of
the fuzzy weight is performed through the possibility transition matrix R
n
, which
results into A
n1
D R
i
n
ıA
n
[
163
,
195
].
