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In-Depth Information
Fig. 10.5
(
a
) input pattern 1 (
b
) recalled pattern 1
It holds
0
1
0
1
1:000 0:0008 0:0037
0:0008 0:9989 0:0042
0:0037 0:0042 1:0794
100
010
001
RR
T
@
A
) RR
T
@
A
D
'
(10.15)
Thus,
matrix R represents
a
unitary
rotation
from
the
vector
space
v
. W
1
1
;
v
. W
1
2
;
v
. W
1
3
to the vector space
v
. W
2
1
;
v
. W
2
2
;
v
. W
2
3
.
3. Simulation tests
Simulation tests have been carried out for the recognition of images stored in
an associative memory. Two binary images were considered (see Figs.
10.5
b and
10.7
b). Both images consisted of 8 8 pixels. The images were selected, so as to
reduce cross pattern interference [
180
].
The correlation weight matrix W 2 R
6464
was found. The images of Figs.
10.5
a,
10.6
a,
10.7
a,
10.8
a were given as input to the weight matrix W . The recalled patterns
are given in Fig.
10.5
b,
10.6
b,
10.7
b,
10.8
b, respectively. This simulation test
demonstrates the known error-correcting capability of neural associative memories.
When the randomly distorted patterns were presented to the associative memory
then after a number of iterations the associative memory converged to the stored
images.
Next, the weight matrix W was viewed as an associative memory with stochastic
weights. The superimposed weight matrices W
i
;iD 1;2; ;2
64
were considered
and the associated rotation matrices R
i
's were calculated. The decomposition of
W into the superposition of weight matrices W
i
followed the analysis presented
in Sect.
10.2.1
. The calculation of the rotation matrices R
i
followed the analysis
presented in Sect.
10.2.2
. The patterns of Figs.
10.9
a and
10.10
a were generated
by applying the stored images of Figs.
10.5
b and
10.7
b to two different rotation
matrices R
i
, respectively. Then by applying the patterns of Figs.
10.9
a and
10.10
a
as input to the matrices R
i
recall of the initial patterns could be observed. This
simulation test demonstrates that the rotation matrices R
i
result in 2
64
different
perceptions of the initial memory patterns.