Digital Signal Processing Reference
In-Depth Information
where
X
1
,
X
2
are tangent vectors at the identity, i.e. two symmetric matrices.
If
Y
1
,
Y
2
are two tangent vectors at
P
∈
P
+
(
n
)
, the group action (
3.4
) with
P
−
1
/
2
transports them to the tangent space at the identity, where
P
−
1
/
2
is
defined as the symmetric square root of
P
. The invariance of the metric then implies
A
=
Tr
P
−
1
Y
1
P
−
1
Y
2
P
−
1
/
2
Y
1
P
−
1
/
2
P
−
1
/
2
Y
2
P
−
1
/
2
g
P
(
Y
1
,
Y
2
)
=
g
I
(
,
)
=
The associated Riemannian distance is
1
/
2
n
log
2
d
P
+
(
n
)
(
P
,
Q
)
=
(λ
i
)
1
λ
1
,...,λ
n
are the eigenvalues of
PQ
−
1
. The main property of this metric is
its invariance to conjugacy and inversion. For any
A
where
∈
GL
(
n
)
and
P
,
Q
∈
P
+
(
n
)
APA
,
AQA
)
=
P
−
1
Q
−
1
d
P
+
(
n
)
(
d
P
+
(
n
)
(
P
,
Q
)
=
d
P
+
(
n
)
(
,
)
Note that, this metric coincides with the Fisher Information Metric (FIM) for the
following statistical inference problem: the available observations have a Gaussian
distribution with zero mean and a covariance matrix parametrized by an unknown
matrix in
P
. The distance distorts the space to measure the amount of
information between the distributions. For instance
d
P
+
(
n
)
(
∈
P
+
(
n
)
P
,
P
+
δ
P
)
→∞
when
0. This is understandable from an information point of view,
as a Gaussian distribution with 0 mean and 0 covariance matrix, carries infinitely
more information than a Gaussian with a strictly positive covariance matrix. Finally
the FIM enjoys invariance properties to reparameterization i.e.
x
P
→
0, for any
δ
P
Ax
in the case
of 0 mean Gaussian distributions, and thus it is no surprise to see it coincide with the
natural metric of the cone.
→
3.2.2 Contraction Property
d
The notion of contraction [
15
] for a dynamical system described by the flow
dt
x
=
can be interpreted as the (exponential) decrease of the (geodesic) distance
between two arbitrary points under the flow.
f
(
x
,
t
)
d
be a smooth dynamical system, defined on a
C
1
Definition 1
Let
dt
x
=
f
(
x
,
t
)
embedded manifold
M
equipped with a Riemanian metric denoted by
g
x
(v
1
,v
2
)
on the tangent space at
x
.Let
X
∈ R
(
x
,
t
)
denote the flow associated to
f
:
d
dt
X
X
(
x
,
0
)
=
x
,
(
x
,
t
)
=
f
(
X
(
x
,
t
),
t
)
(3.5)
