5 Invariant Koszul Form of Homogeneous Bounded Domains …
105
We can observe that we are able to recover the Fisher metric for multivariate
gaussian densities (M, R) with R the covariance matrix, in the case where the mean
vector M = 0. If we use previous metric in the Siegel upper half space:
X = 0, Y = R ⇒ Z = i R ⇒ ds
2
= T race
R
−1 d R
2
(5.25)
In the Siegel unit disk S H Siegel =
W/W W
+
< I d
, extension of Poincaré unit
disk, obtained by Cayley transform W = (Z − i I d )(Z + i I d )
−1 , Siegel metric is
given by (Fig. 5.5):
ds
2
= T race
1 − W ¯
W
dW
1 − ¯
W W
d ¯
W
(5.26)
In next chapter, we will consider the Poincaré unit disk as an Homogeneous
Symplectic Manifolds associated to SU(1,1) co-adjoint orbits by Kirillov-KostantSouriau 2-form. We will then deduced how to define Gaussian density parametrized
in Poincaré unit disk using Jean-Marie Souriau Lie Groups Thermodynamics.
Fig. 5.5 Siegel Upper-Half Space, its metric and distance along geodesics
105
We can observe that we are able to recover the Fisher metric for multivariate
gaussian densities (M, R) with R the covariance matrix, in the case where the mean
vector M = 0. If we use previous metric in the Siegel upper half space:
X = 0, Y = R ⇒ Z = i R ⇒ ds
2
= T race
R
−1 d R
2
(5.25)
In the Siegel unit disk S H Siegel =
W/W W
+
< I d
, extension of Poincaré unit
disk, obtained by Cayley transform W = (Z − i I d )(Z + i I d )
−1 , Siegel metric is
given by (Fig. 5.5):
ds
2
= T race
1 − W ¯
W
dW
1 − ¯
W W
d ¯
W
(5.26)
In next chapter, we will consider the Poincaré unit disk as an Homogeneous
Symplectic Manifolds associated to SU(1,1) co-adjoint orbits by Kirillov-KostantSouriau 2-form. We will then deduced how to define Gaussian density parametrized
in Poincaré unit disk using Jean-Marie Souriau Lie Groups Thermodynamics.
Fig. 5.5 Siegel Upper-Half Space, its metric and distance along geodesics