162
F. Nielsen
Since the Christoffel symbols
i
jk (θ ) = 0 (and
∗i
jk (η) = 0) in a dually flat space,
we have ˙
v
i
= 0 and recover the equation of the primal geodesic (and ˙
v i = 0 for the
equation of the dual geodesic, respectively).
The dual Riemannian metrics [g i j ] = [g ji ] and [g
∗i j
] = [g
∗ ji
] are induced by
the Hessians of the dual potential functions F and F
∗ (both symmetric positivedefinite matrices), respectively. At any given point p ∈ M, we consider the natural
basis {e i = ∂ i =
∂
∂θ i } and the reciprocal basis {e
∗i
= ∂
i
=
∂
∂η i
} i of the tangent plane
T p so that g(e i , e
∗ j
) = δ
j
i . That is, the basis vectors of the primal and reciprocal
basis are mutually orthogonal. We have [g i j ] = [g(e i , e j )] i j = ∇
2 F(θ ) and [g
∗i j
] =
[g
∗
(e
∗i
, e
∗ j
)] i j = ∇
2 F
∗
(η). We can check that the natural primal basis and the dual
reciprocal basis are mutually orthogonal from Crouzeix identity:
∇
2 F(θ )∇
2 F
∗
(η) = I,
(7.12)
where I denotes the D × D identity matrix. Note that Eq. 7.12 can be read using a
single coordinate system as ∇
2 F(θ )∇
2 F
∗
(η(θ )) = ∇
2 F(θ )∇
2 F
∗
(∇ F(θ )) = I or
∇
2 F(θ (η))∇
2 F
∗
(η) = ∇
2 F(∇ F
∗
(η))∇
2 F
∗
(η) = I .
A tangent vector v pq of the primal geodesic γ pq at p is a vector of the tangent plane
T p with contravariant components θ(q) − θ( p) (say, expressed in the primal basis
B p = {e 1 , . . . , e D }): v =
D
i=1 (θ
i
(q) − θ
i
( p))e i with v
i
= θ
i
(q) − θ
i
( p). Indeed,
we have
dγ pq (t)
dt
=
d
dt
((1 − t)θ ( p) + tθ(q)) =
i
(θ
i
(q) − θ
i
( p))∂ i ,
(7.13)
where ∂ i =
∂
∂θ i . Similarly, a tangent vector v
∗
pq at the dual geodesic γ
∗
pq is a vector
of the tangent plane T p with covariant components η(q) − η( p) (expressed in the
reciprocal basis B
∗
p = {e
∗ 1, . . . , e
∗ D
}): v =
D
i=1 (η i i(q) − η i ( p))e
∗i . Indeed, we
check that
dγ
∗
pq (t)
dt
=
d
dt
((1 − t)η( p) + tη(q)) =
i
(η i (q) − η i ( p))∂
i
,
(7.14)
where ∂
i
=
∂
∂η i
. In general, for separable Bregman generators, i.e., F(θ )=
D
i=1 F i (θ
i
)
where the F i ’s are scalar strictly convex and C
3 functions, we can choose both
the primal and reciprocal basis to be orthogonal but they are not necessarily
orthonormal (except in the special case of the Euclidean geometry obtained by
F(θ ) =
D
i=1 sqr(θ
i
) where sqr(x) := x
2 denotes the square function).
The metric tensor field g defines a smooth scalar product g(·, ·) on the tangent
bundle T M (informally, the union of all tangent planes) such that for any two vectors u, v ∈ T p , we have g p (u, v) = u i v
i
= u
i
v i . In each tangent plane, we thus have
an inner product space. The contravariant and covariant components of a vector v
can be retrieved using the inner product with the reciprocal basis and the primal
F. Nielsen
Since the Christoffel symbols
i
jk (θ ) = 0 (and
∗i
jk (η) = 0) in a dually flat space,
we have ˙
v
i
= 0 and recover the equation of the primal geodesic (and ˙
v i = 0 for the
equation of the dual geodesic, respectively).
The dual Riemannian metrics [g i j ] = [g ji ] and [g
∗i j
] = [g
∗ ji
] are induced by
the Hessians of the dual potential functions F and F
∗ (both symmetric positivedefinite matrices), respectively. At any given point p ∈ M, we consider the natural
basis {e i = ∂ i =
∂
∂θ i } and the reciprocal basis {e
∗i
= ∂
i
=
∂
∂η i
} i of the tangent plane
T p so that g(e i , e
∗ j
) = δ
j
i . That is, the basis vectors of the primal and reciprocal
basis are mutually orthogonal. We have [g i j ] = [g(e i , e j )] i j = ∇
2 F(θ ) and [g
∗i j
] =
[g
∗
(e
∗i
, e
∗ j
)] i j = ∇
2 F
∗
(η). We can check that the natural primal basis and the dual
reciprocal basis are mutually orthogonal from Crouzeix identity:
∇
2 F(θ )∇
2 F
∗
(η) = I,
(7.12)
where I denotes the D × D identity matrix. Note that Eq. 7.12 can be read using a
single coordinate system as ∇
2 F(θ )∇
2 F
∗
(η(θ )) = ∇
2 F(θ )∇
2 F
∗
(∇ F(θ )) = I or
∇
2 F(θ (η))∇
2 F
∗
(η) = ∇
2 F(∇ F
∗
(η))∇
2 F
∗
(η) = I .
A tangent vector v pq of the primal geodesic γ pq at p is a vector of the tangent plane
T p with contravariant components θ(q) − θ( p) (say, expressed in the primal basis
B p = {e 1 , . . . , e D }): v =
D
i=1 (θ
i
(q) − θ
i
( p))e i with v
i
= θ
i
(q) − θ
i
( p). Indeed,
we have
dγ pq (t)
dt
=
d
dt
((1 − t)θ ( p) + tθ(q)) =
i
(θ
i
(q) − θ
i
( p))∂ i ,
(7.13)
where ∂ i =
∂
∂θ i . Similarly, a tangent vector v
∗
pq at the dual geodesic γ
∗
pq is a vector
of the tangent plane T p with covariant components η(q) − η( p) (expressed in the
reciprocal basis B
∗
p = {e
∗ 1, . . . , e
∗ D
}): v =
D
i=1 (η i i(q) − η i ( p))e
∗i . Indeed, we
check that
dγ
∗
pq (t)
dt
=
d
dt
((1 − t)η( p) + tη(q)) =
i
(η i (q) − η i ( p))∂
i
,
(7.14)
where ∂
i
=
∂
∂η i
. In general, for separable Bregman generators, i.e., F(θ )=
D
i=1 F i (θ
i
)
where the F i ’s are scalar strictly convex and C
3 functions, we can choose both
the primal and reciprocal basis to be orthogonal but they are not necessarily
orthonormal (except in the special case of the Euclidean geometry obtained by
F(θ ) =
D
i=1 sqr(θ
i
) where sqr(x) := x
2 denotes the square function).
The metric tensor field g defines a smooth scalar product g(·, ·) on the tangent
bundle T M (informally, the union of all tangent planes) such that for any two vectors u, v ∈ T p , we have g p (u, v) = u i v
i
= u
i
v i . In each tangent plane, we thus have
an inner product space. The contravariant and covariant components of a vector v
can be retrieved using the inner product with the reciprocal basis and the primal