Digital Signal Processing Reference
In-Depth Information
m
/
m
1
|||
G
(
A
1
,...,
A
m
)
||| ≤
1
|||
A
j
|||
.
(2.40)
j
=
Special cases of (
2.39
) and (
2.40
) were proved earlier by Yamazaki [
30
]. It turns out
that both the means
G
#
and
G
considered in Sects.
2.4
and
2.5
also satisfy (
2.39
) and
(
2.40
).
It is of interest to know what properties characterise the Riemannian mean
G
among all means. One such property has been found in [
31
] and [
22
]. In the first of
these papers, Yamazaki showed that
m
log
A
j
≤
0
implies
G
(
A
1
,...,
A
m
)
≤
I
.
(2.41)
j
=
1
In [
22
] Lim and Palfia show that this condition together with congruence invariance
and self-duality (conditions 5 and 8 in the ALM list) uniquely determine the mean
G
.
To see this consider any function
g
taking positive matrix values. If
it satisfies the condition (
2.41
) and is self-dual, then
j
=
1
(
A
1
,...,
A
m
)
log
A
j
=
0 implies
log
X
−
1
/
2
then we have
j
=
1
g
(
A
1
,...,
A
m
)
=
I
.
If
X
=
G
(
A
1
,...,
A
m
),
A
j
X
−
1
/
2
=
0
.
Hence
g
X
−
1
/
2
A
1
X
−
1
/
2
X
−
1
/
2
A
m
X
−
1
/
2
,...,
=
I
.
If
g
is congruence-invariant, then from this it follows that
g
(
A
1
,...,
A
m
)
=
X
=
G
(
A
1
,...,
A
m
).
2.7 Summary
The Riemannian mean, also called the Cartan mean or the Karcher mean, has long
been of interest in differential geometry. Recently it has been used in several areas
like radar and medical imaging, elasticity, machine learning and statistics. It is also
an interesting topic for matrix analysts and operator theorists. Some questions (like
operator monotonicity and concavity) that are intrinsically more natural to these
subjects have led to a better understanding of this object. In particular several new
characterisations of the Riemannian mean have been found in 2010-2011. These
show the mean as a limit of (explicitly constructed) sequences. They will be useful
for devising numerical algorithms for computation of the mean.
