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λ
min
>
1
2
2
w
λ
min
<
1
1
O
t
0
t
∞
t
Figure 2.6
Asymptotic behavior of the OJA
+
ODE for different values (circles) of the initial
conditions.
whose solution is given by
p
0
p
0
+
(
1
−
p
0
)
e
2
(λ
n
−
1
)(
t
−
t
0
)
p
=
if
p
0
=
1
(2.121)
The change in the time constant of the exponential with respect to OJA implies
a fundamental difference. Indeed, if
λ
n
>
1, OJAn behaves as OJA (the OJA
expressions are valid here simply by replacing
λ
n
with
λ
n
−
1). If
λ
n
=
1,
p
=
p
0
.
If
λ
n
<
1,
p
→
1, as expected from Theorem 48. Figure 2.6 summarizes the
results for different values of
p
0
.
In summary: OJA
+
is not divergent and it does not suffer from the sudden
divergence, but, unfortunately, it requires the assumption that the smallest eigen-
value of the autocorrelation matrix of the input data is less than unity. If it cannot
be assumed in advance (e.g., for noisy data), OJA
+
may suffer from the sudden
divergence.
2.6.2.4 Simulation Results for the MCA Divergence
The previous analy-
sis is illustrated by using, as a benchmark, the example in [124, pp. 294-295] and
[181]. A zero-mean Gaussian random vector
x
(
t
)
is generated with the covariance
matrix
#
%
0
.
4035
0
.
2125
0
.
0954
$
&
R
=
0
.
2125
0
.
3703
0
.
2216
(2.122)
0
.
0954
0
.
2216
0
.
4159
and taken as an input vector. The learning rate is given by
α(
t
)
=
const
=
0
.
01.
The algorithms are initialized at the true solution; that is,
4473]
T
w(
0
)
=
[0
.
4389
−
0
.
7793 0
.
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