Digital Signal Processing Reference
In-Depth Information
and
u
ij
(
P
)
¼
Tr{
S
ij
(
PP
)} for 1
i j n
. The relevance of this basis is shown
by the following relation proved in [24, 25]
P ¼ P
þ
X
(
i
,
j
)
[
P
s
2
u
ij
(
P
)
S
ij
þO
(
kPP
k
Fro
)
(4
:
80)
def
{(
i
,
j
)
j
1
i j n
and
i r
}. There are
2
(2
nr þ
1) pairs in
P
s
and
this is exactly the dimension of the manifold of the
n n
rank-
r
symmetric matrices.
This point, together with relation (4.80), shows that the matrix set
fS
ij
j
(
i
,
j
)
[
P
s
g
is in
fact an orthonormal basis of the tangent plane to this manifold at point
P
. In other
words, a
n n
rank-
r
symmetric matrix
P
lying less than
1
away from
P
(i.e.,
kP
2
P
k
,
1
) has negligible (of order
1
2
) components in the direction of
S
ij
for
r
,
i j n
. It follows that, in a neighborhood of
P
, the
n n
rank-
r
symmetric
matrices are uniquely determined by the
2
(2
nr þ
1)
1 vector
u
(
P
) defined by:
u
(
P
)
¼
where
P
s
¼
def
T
vec(
PP
), where
S
denotes the following
n
2
r
S
2
(2
nr þ
1) matrix:
def
[
...
, vec(
S
ij
),
...
], (
i
,
j
)
[
P
s
.If
P
(
u
) denotes the unique (for
kuk
sufficiently
small)
n n
rank-
r
symmetric matrix such that
S
S¼
T
vec(
P
(
u
)
P
)
¼ u
, the following
one-to-one mapping is exhibited for sufficiently small
ku
(
k
)
k
2
)
!u
(
k
)
vec{
P
[
u
(
k
)]}
¼
vec(
P
)
þSu
(
k
)
þO
(
ku
(
k
)
k
T
vec[
P
(
k
)
P
]
:
¼S
(4
:
81)
We are now in a position to solve the Lyapunov equation in the new parameter
u
.
The stochastic equation governing the evolution of
u
(
k
) is obtained by applying
the transformation
P
(
k
)
!u
(
k
)
¼S
T
vec[
P
(
k
)
P
] to the original equation (4.79),
thereby giving
u
(
k þ
1)
¼ u
(
k
)
þm
k
f
[
u
(
k
),
x
(
k
)]
þm
k
c
[
u
(
k
),
x
(
k
)]
(4
:
82)
def
def
T
vec[
h
(
P
(
u
),
xx
T
)].
Solving now the Lyapunov equation associated with (4.82) after deriving
the derivative of the mean field
f
(
u
) and the covariance of the field
f
[
u
(
k
),
x
(
k
)]
for independent Gaussian distributed data
x
(
k
), yields the covariance
C
u
of the asymp-
totic distribution of
u
(
k
). Finally using mapping (4.81), the covariance
C
P
¼ SC
u
S
T
vec[
f
(
P
(
u
),
xx
T
)] and
c
(
u
,
x
)
¼
where
f
(
u
,
x
)
¼
S
S
T
of
the asymptotic distribution of
P
(
k
) is deduced [25]
X
l
i
l
j
2(
l
i
l
j
)
(
u
i
u
j
þu
j
u
i
)(
u
i
u
j
þu
j
u
i
)
T
C
P
¼
:
(4
:
83)
1
ir
,
jn
To improve the learning speed and misadjustment tradeoff of Oja's algorithm (4.30),
it has been proposed in [25] to use the recursive estimate (4.20) for
C
x
(
k
)
¼
E[
x
(
k
)
x
T
(
k
)]. Thus the modified Oja's algorithm, called the smoothed Oja's
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