Digital Signal Processing Reference
In-Depth Information
Following the same approach with now
W
0
(
k þ
1)
¼ W
(
k
)
+x
(
k
)
x
T
(
k
)
W
(
k
)
G
(
k
) where
G
(
k
)
¼ m
k
Diag(1,
a
2
,
...
,
a
r
), prove that
"
#
w
j
(
k
)
w
j
(
k
)
w
i
(
k þ
1)
¼ w
i
(
k
)
þa
i
m
k
I
n
w
i
(
k
)
w
i
(
k
)
X
i
1
a
j
a
i
1
þ
j¼
1
x
(
k
)
x
T
(
k
)
w
i
(
k
)
þO
(
m
k
)
for
i ¼
1,
...
,
r:
4.14
Specify the stationary points of the ODE associated with algorithm (4.58).
Using the eigenvalues of the derivative of the mean field of this algorithm,
prove that if
l
n
2
rþ
1
,
1
and
b
.
l
n
r
þ
1
l
n
1, the only asymptotically stable
points of the associated ODE are the eigenvectors
+v
nrþ
1
,
...
,
+v
n
.
4.15
Prove that the set of the
n r
orthogonal matrices
W
(denoted the Stiefel mani-
fold
St
n,r
) is given by the set of matrices of the form
e
A
W
where
W
is an arbi-
trary
n r
fixed orthogonal matrix and
A
is a skew-symmetric matrix
(
A
T
¼
2
A
).
Prove the following relation
J
(
WþdW
)
¼ J
(
W
)
þ
Tr[
dA
T
(
H
2
WH
1
W
T
WH
1
W
T
H
2
)]
þo
(
dW
)
where
J
(
W
)
¼
Tr[
WH
1
W
T
H
2
] (where
H
1
and
H
2
are arbitrary
r r
and
n n
symmetric matrices) defined on the set of
n r
orthogonal matrices. Then, give
the differential
dJ
of the cost function
J
(
W
) and deduce the gradient of
J
(
W
)on
this set of
n r
orthogonal matrices
7
W
J ¼
[
H
2
WH
1
W
T
WH
1
W
T
H
2
]
W:
(4
:
93)
4.16
Prove that if
¯
(
u
)
¼
2
7
u
J
, where
J
(
u
) is a positive scalar function,
J
[
u
(
t
)] tends
to a constant as
t
tends to
1
, and consequently all the trajectories of the ODE
(4.69) converge to the set of the stationary points of the ODE.
4.17
Let
u
be a stationary point of the ODE (4.69). Consider a Taylor series expan-
sion of
¯
(
u
) about the point
u ¼ u
df
(
u
)
du
ju¼u
f
(
u
)
¼ f
(
u
)
þ
(
uu
)
þO
[(
uu
)](
uu
)
:
By admitting that the behavior of the trajectory
u
(
t
) of the ODE (4.69) in the
neighborhood of
u
is identical to those of the associated linearized ODE
d
u
(
t
)
dt
¼ D
[
u
(
t
)
u
] with
D¼
about the point
u
, relate the
stability of the stationary point
u
to the behavior of the eigenvalues of the
matrix
D
.
def
df
(
u
)
du
ju¼u
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