Digital Signal Processing Reference
In-Depth Information
Concavity of the geometric mean.
In the preceding discussions we found that
the geometric mean of two positive numbers is concave in the positive or-
thant
x
i
>
0
.
Similarly it is true that the geometric mean of
P
numbers
g
(
x
0
,x
1
,...,x
P−
1
)=
x
k
1
/P
P−
1
(21
.
21)
k
=0
is concave in the positive orthant
x
k
>
0
.
To see this we first compute the
Hessian matrix, which turns out to be
G
=
c
H
,
where
c
=(
P−
1
k
=0
x
k
)
1
/P
/P
2
>
0
,
and
⎡
⎤
y
0
0
...
0
y
1
0
...
0
⎣
⎦
T
H
=
yy
−
P
,
.
.
.
.
.
.
y
P−
1
00
...
where
y
=[
y
0
y
1
... y
P−
1
]
T
,
is a real column vector with
y
k
=1
/x
k
.
To prove concavity of the geometric mean, it therefore su
ces to prove that
the real Hermitian matrix
H
is negative semidefinite, that is,
v
T
Hv
≤
0for
all real
v
.Now,
⎡
⎣
⎤
⎦
v
y
0
0
...
0
0
y
1
...
0
T
T
T
T
v
Hv
=(
v
y
)(
y
v
)
−
P
v
.
.
.
.
.
.
y
P−
1
00
...
which can be rewritten as
Hv
=
k
v
k
y
k
2
P
k
T
v
k
y
k
v
−
(21
.
22)
v
P−
1
y
P−
1
]
T
,
z
=[1 1
...
1]
T
,
Defining the real vectors
u
=[
v
0
y
0
...
and applying the Cauchy-Schwartz inequality, we get, (
u
T
z
)
2
≤
(
u
T
u
)(
z
T
z
)
,
that is,
v
k
y
k
2
P
k
v
k
y
k
.
≤
k
T
Using this in Eq. (21.22) shows that
v
Hv
≤
0 for all real
v
.So
H
is
negative semidefinite, proving that the geometric mean is a concave function
in the positive orthant
x
k
>
0
.
21.2.5 Composite functions
If
f
(
x
)and
h
(
x
) are convex, can we say
h
(
f
(
x
)) is convex as well? If
h
(
x
)is
an
increasing
function, this is indeed true. More precisely we have the following
(see Marcus and Minc [1964] p. 102):
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