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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.
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