Digital Signal Processing Reference
In-Depth Information
Fig. 9.3
Fréchet-Karcher flow on Cartan-Hadamard manifold
m
n
+
1
=
γ
n
(
)
=
exp
m
n
(
−
·∇
(
m
n
)
)
γ
n
(
)
=−∇
(
m
n
)
t
t
f
with
0
f
(9.26)
In the discrete case, the center of mass for finite set of points is given by:
t
M
exp
−
1
m
n
+
1
=
exp
m
n
·
m
n
(
x
i
)
(9.27)
i
=
1
Fréchet, inventor of Cramer-Rao bound in 1939 (published in Institut Henri Poincaré
Lecture of Winter 1939 on statistics) [
53
], has also introduced the entire concept of
Metric Spaces Geometry and functional theory on this space (any normed vector
space is a metric space by defining
d
but not the contrary). On this
base, Fréchet has then extended probability in abstract spaces (Fig.
9.3
).
A different point of view on center of mass or barycenter has been followed by
Emery [
48
]. He has defined the expectation
E
[
x
] as the set of all
x
such that:
(
x
,
y
)
=
y
−
x
ψ(
x
)
≤
E
[
ψ(
x
)
]
(9.28)
for all continuous convex functions. A related point of view was used by Doss
and Herer who define
E
[
x
] to be the set of all
x
such that:
d
(
z
,
x
)
≤
E
[
d
(
z
,
X
)
]
(9.29)
In this framework, expectation
b
=
E
[
g
(
x
)
] of an abstract probabilistic variable
g
where
x
lies on a manifold is introduced by Emery [
48
] as an exponential
barycenter:
(
x
)
exp
−
1
b
(
g
(
x
))
P
(
dx
)
=
0
(9.30)
M
[·]
In Classical Euclidean space, we recover classical definition of Expectation E
:
