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are, respectively,
)
+
w
)
w (
d
w (
t
)
dt
T
=−
R
w (
t
(
t
)
R
w (
t
t
)
(2.20)
T
d
w (
t
)
)
+
w
(
t
)
R
w (
t
)
=−
R
w (
t
w (
t
)
(2.21)
w
T
(
t
)
w
(
t
)
dt
Rayleigh quotient
The solutions
w (
t
)
of eq. (2.16) [eq. (2.17)] tend to the asymptotically stable
points of eq. (2.20) [eq. (2.21)].
In [195], the following theorem is presented.
Theorem 47 (Asymptotic Stability)
Let R be positive semidefinite with min-
imum eigenvalue
λ
n
of multiplicity 1 and corresponding unit eigenvector z
n
.If
w
T
(
0
)
z
n
=
0
, then:
1.
For eq.
(
2.20
)
, asymptotically
w (
t
)
is parallel to z
n
.
2.
In the special case of
λ
n
=
0
, it holds that
lim
t
→∞
w (
t
)
=
w
(
0
)
z
n
z
n
.
T
3.
For eq.
(
2.21
)
, it holds that
lim
t
→∞
w (
t
)
=±
z
n
and
T
t
→∞
w
lim
(
t
)
R
w (
t
)
=
λ
n
(2.22)
Proof.
See [195, App., p. 455; 142, pp. 72-75].
Another Oja's learning rule [141] (
OJA
+
) is the following:
w (
t
+
1
)
=
w (
t
)
−
α (
t
)
y
(
t
)
x
(
t
)
−
(
y
2
(
t
)
2
2
+
1
−
w (
t
)
) w (
t
)
]
(2.23)
The corresponding averaging ODE is given by
)
+
w
)
w (
d
w (
t
)
T
=−
R
w (
t
(
t
)
R
w (
t
t
)
dt
2
2
+
w (
t
)
−
w (
t
)
w (
t
)
(2.24)
The following theorem is also given.
Theorem 48 (Asymptotic Stability)
In eq.
(
2.23
)
assume that the eigenval-
ues of R satisfy
λ
1
> λ
2
>
···
> λ
n
>
0
and
λ
n
<
1
and that the learning rate
α (
t
)
is positive. Then
lim
t
→∞
w (
t
)
=
z
min
(
or
−
z
min
)
if
w
T
(
0
)
z
min
is positive
(
or negative
)
.
Proof.
See [141, App., p. 935].
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