Digital Signal Processing Reference
In-Depth Information
the arithmetic mean is compatible with the algebraic structure of vector spaces. In
the case of groups, the compatibility with the group structure requires the invariance
with respect to left and right multiplications (the group can be non-commutative)
and the inversion operator. When we translate (or inverse) a given set of samples
or a probability measure, it is reasonable to desire that their mean be translated (or
inversed) exactly in the same way.
Since one-parameters subgroups and their relationship with the Lie algebra are
key properties in a Lie group, one could expect to define a mean using these features.
For instance, the Log-Euclidean mean proposed in [
7
] maps the data points
for
which the logarithm is well defined to the Lie algebra; takes the Euclidean mean with
(non-negative and normalized) weights
{
x
i
}
w
i
and exponentiates the result:
Exp
i
x
LE
=
¯
w
i
Log
(
x
i
)
.
(7.3)
h
(
−
1
)
is the log-Euclidean
This definition is consistent with conjugation as
h
·¯
x
LE
·
h
(
−
1
)
thanks to Theorem 7.6. However, this definition fails
to be invariant under left and right translation!
A well-established approach to define a notion of mean compatible with algebraic
operations is to define first a
distance
(or metric) compatible with these operations
and then to rely on this distance to define the mean. Indeed, one can generalize the
classical notion of arithmetic mean by relying on the minimal variance or dispersion
in the general setting of metric spaces [
27
]: the Fréchet mean (also called Riemannian
center of mass) of the data points
mean of the points
h
·
x
i
·
{
x
i
}
with the non-negative weights
w
i
is the set of
points:
)
α
.
arg min
y
w
i
·
dist
(
x
i
,
y
(7.4)
∈
E
i
The case
α
=
1to
the median. The existence and uniqueness of these means on Riemannian manifolds
has been studied first by Karcher (who relax the definition to local minima) [
37
] and
then in [
3
,
38
,
44
,
45
,
68
,
69
]. Thus, it seems natural to investigate if we can define
a Riemannian metric compatible with the Lie group operations.
2 corresponds in vector spaces to the arithmetic mean, the case
α
=
7.3.1 Bi-Invariant Metrics on Lie Groups
A Riemannian metric is a smooth collection of positive definite bilinear forms on
tangent spaces of the manifold. In the case of Lie groups, we can require the metric
to be left-invariant (invariant by the left translation), or right-invariant. The left-
invariance requires that for any two points
g
and
h
of
G
and any vectors
v
and
w
of
T
g
G
. In other words, a metric
is left-invariant if all left translations are isometries. It is easy to see that all left-
invariant metrics on a Lie group are determined by the inner product at the identity:
,wehave:
<
DL
h
v,
DL
h
w>
T
h
·
g
G
=
<v,w>
T
g
G
