7 On Geodesic Triangles with Right Angles in a Dually Flat Space
163
basis, respectively: v
i
= g(v, e
∗i
), and v i = g(v, e i ). We can also use vector-matrix
multiplications of linear algebra to calculate the inner product: g p (u, v) = [u
i
]
×
∇
2 F(θ ( p)) × [v
i
] = [u i ]
× ∇
2 F
∗
(η( p)) × [v i ], where [u
i
] and [u i ] denote the
column-vector of the contravariant components of u and covariant components of u,
respectively. Since v
i
= ∇
2 F
∗
(η( p)) × [v i ] and v i = ∇
2 F(θ ( p)) × [v
i
], we therefore have the following equivalent rewritings of the inner product:
g p (u, v) = [u i ]
× [v
i
],
(7.15)
= [u
i
]
× [v j ],
(7.16)
= [u
i
]
× ∇
2 F(θ ( p)) × [v
i
],
(7.17)
= [u i ]
× ∇
2 F
∗
(η( p)) × [v i ].
(7.18)
When the generator F(θ ) is separable, i.e., F(θ ) =
D
i=1 F i (θ
i
) for univariate
generators F i ’s, the inner product at a tangent plane T p writes equivalently as (using
Einstein summation convention):
g p (u, v) = u
i F
i (θ
i
( p))v
i
,
(7.19)
= u i F
∗
i
(η i ( p))v i ,
(7.20)
= u i v
i
,
(7.21)
= u
i
v i .
(7.22)
Given two vectors u, v ∈ T p , we measure their lengths u p and v p and the interior
angle α p (u, v) between them as:
u p =
g p (u, u) =
u i u i ,
(7.23)
v p =
g p (v, v) =
v i v i ,
(7.24)
α p (u, v) = arccos
u
i
v i
u p v p
= α p (v, u).
(7.25)
Notice that the Hessian metric g and g
∗ are not conformal (i.e., not a positive scalar
function of the Euclidean metric), and that we cannot “read” the angles directly
in the θ - or η-coordinate systems. In other words, the Euclidean angles displayed
in the θ - or η-coordinate systems do not correspond to the intrinsic angles of the
underlying Bregman geometry. Figure 7.5 shows a visualization of the metric tensor
field for the Itakura–Saito manifold (with [g i j ](θ ) = ∇
2 F(θ ) = diag(
1
sqr(θ 1 ,
1
sqr(θ 2 )
)
for 2D θ = (θ
1
, θ
2
)). Dually flat spaces are a particular case of more general (local)
Hessian structures studied in [29].
In information geometry [3], the dually flat torsion-free affine connections
4
∇ and
∇
∗ are coupled the metric tensor g. An affine connection ∇ is flat iff. there exists
4 The notion of dual connections of information geometry is more general than the notion of conjugate connections of affine differential geometry [10] which stems from dual affine immersions.
163
basis, respectively: v
i
= g(v, e
∗i
), and v i = g(v, e i ). We can also use vector-matrix
multiplications of linear algebra to calculate the inner product: g p (u, v) = [u
i
]
×
∇
2 F(θ ( p)) × [v
i
] = [u i ]
× ∇
2 F
∗
(η( p)) × [v i ], where [u
i
] and [u i ] denote the
column-vector of the contravariant components of u and covariant components of u,
respectively. Since v
i
= ∇
2 F
∗
(η( p)) × [v i ] and v i = ∇
2 F(θ ( p)) × [v
i
], we therefore have the following equivalent rewritings of the inner product:
g p (u, v) = [u i ]
× [v
i
],
(7.15)
= [u
i
]
× [v j ],
(7.16)
= [u
i
]
× ∇
2 F(θ ( p)) × [v
i
],
(7.17)
= [u i ]
× ∇
2 F
∗
(η( p)) × [v i ].
(7.18)
When the generator F(θ ) is separable, i.e., F(θ ) =
D
i=1 F i (θ
i
) for univariate
generators F i ’s, the inner product at a tangent plane T p writes equivalently as (using
Einstein summation convention):
g p (u, v) = u
i F
i (θ
i
( p))v
i
,
(7.19)
= u i F
∗
i
(η i ( p))v i ,
(7.20)
= u i v
i
,
(7.21)
= u
i
v i .
(7.22)
Given two vectors u, v ∈ T p , we measure their lengths u p and v p and the interior
angle α p (u, v) between them as:
u p =
g p (u, u) =
u i u i ,
(7.23)
v p =
g p (v, v) =
v i v i ,
(7.24)
α p (u, v) = arccos
u
i
v i
u p v p
= α p (v, u).
(7.25)
Notice that the Hessian metric g and g
∗ are not conformal (i.e., not a positive scalar
function of the Euclidean metric), and that we cannot “read” the angles directly
in the θ - or η-coordinate systems. In other words, the Euclidean angles displayed
in the θ - or η-coordinate systems do not correspond to the intrinsic angles of the
underlying Bregman geometry. Figure 7.5 shows a visualization of the metric tensor
field for the Itakura–Saito manifold (with [g i j ](θ ) = ∇
2 F(θ ) = diag(
1
sqr(θ 1 ,
1
sqr(θ 2 )
)
for 2D θ = (θ
1
, θ
2
)). Dually flat spaces are a particular case of more general (local)
Hessian structures studied in [29].
In information geometry [3], the dually flat torsion-free affine connections
4
∇ and
∇
∗ are coupled the metric tensor g. An affine connection ∇ is flat iff. there exists
4 The notion of dual connections of information geometry is more general than the notion of conjugate connections of affine differential geometry [10] which stems from dual affine immersions.