7 On Geodesic Triangles with Right Angles in a Dually Flat Space
159
besides the dually flat spaces, Pythagorean theorems with corresponding divergence
identities have also been reported for α-divergences on the probability simplex [3]
and the logarithmic L
α
H -divergences [32] where H is an exponentially concave generator.
The paper is organized as follows: First, we quickly review the construction of
Bregman manifolds in Sect. 7.2, thereby introducing familiar concepts and notations
of information geometry [3], and give as examples of Bregman manifolds details
for the Mahalanobis manifolds (Sect. 7.2.4.1), the extended Kullback–Leibler manifold (Sect. 7.2.4.2), the Itakura–Saito manifold (Sect. 7.2.4.3), and the multinoulli
manifolds (Sect. 7.2.4.4). Then Sect. 7.3 shows, whenever it is possible, how to build
geodesic ∇-triangles which have one, two or three right angles (thus necessarily
exhibiting angle excesses for two and three right angle triangles). In Sect. 7.4, we
prove that given two distinct points p and q, the locii of points r for which we have
simultaneously the dual Pythagorean theorems holding at r are the intersection of
an autoparallel ∇-submanifold with an autoparallel ∇
∗ submanifold (i.e., the intersection of a θ -flat with a η-flat [3]). We report explicitly the construction method
for the 2D Itakura–Saito manifold and visualize several such triangles. Finally, we
summarize and hint at further perspectives in Sect. 7.5.
7.2 Dually Flat Spaces: Bregman Manifolds
We first explain the mutually orthogonal primal basis and reciprocal basis in an
inner product space in Sect. 7.2.1. Then we describe dual geodesics and their tangent
vectors and the dual parallel transport in Sect. 7.2.2. In Sect. 7.2.3, we explain the
Pythagorean theorem, and Sect. 7.2.4 provides some common examples of Bregman
manifold: Mahalanobis self-dual manifolds, the extended Kullback–Leibler manifold
and the Itakura–Saito manifold.
7.2.1 Preliminary: Inner Product Space and Reciprocal Basis
An inner product space is a vector space V equipped with a symmetric positive
definite bilinear form ·, ·· : V × V → R called the inner product. The length of a
vector v ∈ V is given by its induced norm v =
√ v, v, and the angle between any
two vectors u and v is measured as α(u, v) = arccos
u,v
uv
(in radians). We consider finite D-dimensional inner product spaces where a vector v can be expressed
in any basis B = {e 1 , . . . , e D } (a maximum set of linearly independent vectors) by
its components v B = (v
1
, . . . , v
D
): v =
D
i=1 v
i e i . Vector v can also be expressed
equivalently in an other basis ˆ
B = {ˆ e 1 , . . . , ˆ
e D }: v =
D
i=1 ˆ
v
i
ˆ
e i . Notice that in general the components v B and v ˆ
B are different although they express the same geometric
vector v according to their respective basis B and ˆ
B. We can express the basis vectors
159
besides the dually flat spaces, Pythagorean theorems with corresponding divergence
identities have also been reported for α-divergences on the probability simplex [3]
and the logarithmic L
α
H -divergences [32] where H is an exponentially concave generator.
The paper is organized as follows: First, we quickly review the construction of
Bregman manifolds in Sect. 7.2, thereby introducing familiar concepts and notations
of information geometry [3], and give as examples of Bregman manifolds details
for the Mahalanobis manifolds (Sect. 7.2.4.1), the extended Kullback–Leibler manifold (Sect. 7.2.4.2), the Itakura–Saito manifold (Sect. 7.2.4.3), and the multinoulli
manifolds (Sect. 7.2.4.4). Then Sect. 7.3 shows, whenever it is possible, how to build
geodesic ∇-triangles which have one, two or three right angles (thus necessarily
exhibiting angle excesses for two and three right angle triangles). In Sect. 7.4, we
prove that given two distinct points p and q, the locii of points r for which we have
simultaneously the dual Pythagorean theorems holding at r are the intersection of
an autoparallel ∇-submanifold with an autoparallel ∇
∗ submanifold (i.e., the intersection of a θ -flat with a η-flat [3]). We report explicitly the construction method
for the 2D Itakura–Saito manifold and visualize several such triangles. Finally, we
summarize and hint at further perspectives in Sect. 7.5.
7.2 Dually Flat Spaces: Bregman Manifolds
We first explain the mutually orthogonal primal basis and reciprocal basis in an
inner product space in Sect. 7.2.1. Then we describe dual geodesics and their tangent
vectors and the dual parallel transport in Sect. 7.2.2. In Sect. 7.2.3, we explain the
Pythagorean theorem, and Sect. 7.2.4 provides some common examples of Bregman
manifold: Mahalanobis self-dual manifolds, the extended Kullback–Leibler manifold
and the Itakura–Saito manifold.
7.2.1 Preliminary: Inner Product Space and Reciprocal Basis
An inner product space is a vector space V equipped with a symmetric positive
definite bilinear form ·, ·· : V × V → R called the inner product. The length of a
vector v ∈ V is given by its induced norm v =
√ v, v, and the angle between any
two vectors u and v is measured as α(u, v) = arccos
u,v
uv
(in radians). We consider finite D-dimensional inner product spaces where a vector v can be expressed
in any basis B = {e 1 , . . . , e D } (a maximum set of linearly independent vectors) by
its components v B = (v
1
, . . . , v
D
): v =
D
i=1 v
i e i . Vector v can also be expressed
equivalently in an other basis ˆ
B = {ˆ e 1 , . . . , ˆ
e D }: v =
D
i=1 ˆ
v
i
ˆ
e i . Notice that in general the components v B and v ˆ
B are different although they express the same geometric
vector v according to their respective basis B and ˆ
B. We can express the basis vectors