7 On Geodesic Triangles with Right Angles in a Dually Flat Space
173
1
2
θ 1
2
+
1
2
θ 2
2
=
θ 1 − θ 2
2
2
+
θ 1 + θ 2
2
2
,
(7.74)
2θ 1
2
+ 2θ 2
2
= =θ 1 − θ 2
2
+ +θ 1 + θ 2
2
,
(7.75)
the usual parallelogram identity for the L 2 -normed space.
Finally, let us mention some special submanifolds in dually flat spaces: A kdimensional submanifold S ⊂ M of a dually flat manifold M is said ∇-autoparallel [3]
(∇
∗ -autoparallel) iff. the sets of points of S can be described by a k-dimensional
affine flat in the θ -coordinate system (η-coordinate system, respectively). Primal γ pq
geodesics are 1D ∇-autoparallel submanifolds and dual γ
∗
pq geodesics are 1D ∇
∗ -
autoparallel submanifolds. Notice that the intersection of θ -flats is a θ -flat, but the
intersection of a θ -flat with a η-flat is in general neither a θ -flat nor a η-flat.
In a Bregman manifold, we can also express the divergence between two points p
and q using the Fenchel-Young divergence A F (also called the canonical divergence
of dually flat spaces):
D F ( p : q) = A F (θ ( p) : η(q)) := F((θ ( p)) + F
∗
(η(q)) − θ( p)
η(q). (7.76)
Thus we have
D F ( p : q) = B F (θ ( p) : θ(q)) = A F (θ ( p) : η(q)) = A F ∗ (η(q) : θ( p))
= B F∗ (η(q) : η( p)) = D F ∗ (q : p),
(7.77)
and D F ∗ (q : p) = D F
∗
(q : p) = D F ( p : q) (where D F
∗ is the reverse divergence).
7.2.4 Some Examples of Bregman Manifolds
We describe concisely the Mahalanobis manifolds in Sect. 7.2.4.1, the extended
Kullback–Leibler manifold in Sect. 7.2.4.2, the Itakura–Saito manifold in Sect.
7.2.4.3, and the multinoulli manifolds (Sect. 7.2.4.4).
7.2.4.1 The Mahalanobis Manifolds
Consider the case where the Bregman generator is defined by
F Q (θ ) =
1
2
θ
Qθ,
(7.78)
for a prescribed symmetric positive-definite D × D matrix Q 0. The LegendreFenchel convex conjugate [13, 14] is