7 On Geodesic Triangles with Right Angles in a Dually Flat Space
189
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A. Notations
F Strictly convex and C
3 real-valued function
F
∗
Dual Legendre-Fenchel convex conjugate
θ( p) = (θ
1
( p), . . . , θ
D
( p)) primal coordinates of point p
η( p) = (η 1 ( p), . . . , η D ( p)) dual coordinates of point p
θ ab = θ a − θ b notational shortcut
η ab = η a − η b notational shortcut
D F ( p : q) Divergence between points
B F (θ ( p) : θ(q)) Bregman divergence
A F (θ ( p) : η(q)) Fenchel-Young divergence
(M, g, ∇, ∇
∗
) Dually flat space (Bregman manifold)
T p Tangent plane at p
g p (u, v) inner product between two vectors u and v of T p
[g i j =[g(e i , e j )] i j = ∇
2F
(θ )] Riemannian metric
[g
∗i j
=[g
∗
(e
∗i
, e
∗ j
)] i j = ∇
2 F
∗
(η)] dual Riemannian metric
p,q (v) primal parallel transport of v ∈ T p to T q
∗
p,q (v) dual parallel transport of v ∈ T p to T q
γ ab (t) Primal geodesic: θ(γ ab (t)) = (1 − t)θ (a) + tθ(b)
γ ab (t)
∗
Dual geodesic: η(γ
∗
ab (t)) = (1 − t)η(a) + tη(b)
(v) B vector components in basis B, arranged in a D-tuple
[v B ] vector components in basis B, arranged in a D-dimensional column vector
B p = {e i = ∂ i =
∂
∂θ i } natural basis at T p
B
∗
p = {e
∗i
= ∂
i
=
∂
∂η i
} i reciprocal basis at T p so that g(e i , e
∗ j
) = δ
j
i
v ab =
d
dt
γ ab (0) = ˙
γ ab (0) tangent vector of γ ab (t) at a with contravariant components θ(b) − θ(a)
v
∗
ab =
d
dt
γ
∗
ab (0) = ˙
γ
∗
ab (0) tangent vector of γ
∗
ab (t) at a with covariant components
η(b) − η(a)
[v
i
B ] contravariant components of vector v, v
i
= g(v, e
∗i
)
[v i B ] covariant components of vector v (meaning [v] B ∗ ), v i = g(v, e i )
g p (u, v) inner product at T p of two vectors: g p (u, v) = u i v
i
= u
i
v i .
References
1. Akaho, S., Hino, H., Murata, N.: On a convergence property of a geometrical algorithm for
statistical manifolds (2019). arXiv:1909.12644