7 On Geodesic Triangles with Right Angles in a Dually Flat Space
167
In a Bregman manifold, the remarkable following generalized Pythagorean theorem holds:
Theorem 1 (Generalized Pythagorean theorem) When a primal geodesic γ pq is
orthogonal to a dual geodesic γ
∗
qr at point q (i.e., γ pq ⊥ q γ
∗
qr ), we have (θ ( p) −
θ(q))
(η(r ) − η(q)) = 0, and the following Pythagorean divergence identity holds:
D F ( p : q) + D F (q : r ) = D F ( p : r ).
(7.36)
Proof The proof proceeds in two steps:
• First, Let us show that the test (θ ( p) − θ(q))
(η(r ) − η(q)) = 0 can be rewritten
as an inner product showing that
(θ ( p) − θ(q))
(η(r ) − η(q)) = g q (v qp , v
∗
qr ) = g q ( ˙
γ qp (0), ˙
γ
∗
qr (0)), (7.37)
where v qp denotes the tangent vector at q of the primal geodesic γ pq , and v
∗
qr
is the tangent vector at q of the dual geodesic γ
∗
qr . Indeed, the inner product is
g(v qp , v
∗
qr ) = (v qp )
i
(v
∗
qr ) i and we have
(θ ( p) − θ(q))
(η(r ) − η(q)) = [θ pq ]
× ∇
2 F(θ (q))(∇
2 F(θ (q)))
−1
× [η
∗
rq ],
(7.38)
where θ pq := θ( p) − θ(q) (contravariant components of v qp ) and η
∗
rq := η(r ) −
η(q) (covariant components of v
∗
qr ). That is, [(v qp )
i
] = θ pq and [v
∗
qp ] i = η pq .
Using the Crouzeix identity [6] of Eq. 7.12, we get
(θ ( p) − θ(q))
(η(r ) − η(q)) = [θ pq ]
∇
2 F(θ (q))(∇
2 F
∗
(η(q)))[η
∗
rq ]. (7.39)
The term (∇
2 F
∗
(η(q))) × [η
∗
rq ] gives the contravariant components θ
∗
rq of the
vector v
∗
qr . Thus we have checked that
(θ ( p) − θ(q))
(η(r ) − η(q)) = [θ pq ]
× ∇
2 F(θ (q)) × [θ
∗
rq ] = g q (v qp , v
∗
qr ).
(7.40)
That is,
(θ ( p) − θ(q))
(η(r ) − η(q)) = 0 ⇔ ˙
γ pq (0) ⊥ q ˙
γ
∗
qr (0).
(7.41)
• Now, let us prove the Pythagorean identity when γ pq ⊥ q γ
∗
qr . We use the Bregman
3-parameter identity [4] which generalizes the Euclidean law of cosines:
Property 1 (Bregman 3-parameter identity)
B F (θ 1 : θ 2 ) = B F (θ 1 : θ 3 ) + B F (θ 3 : θ 2 ) − (θ 1 − θ 3 )
(∇ F(θ 2 ) − ∇ F(θ 3 )) ≥ 0
(7.42)
Instantiating this identity with θ 1 = θ( p), θ 2 = θ(r ) and θ 3 = θ(q), and plugging
the fact that (θ ( p) − θ(q))
(∇ F(θ (r )) − ∇ F(θ (q))) = 0, we get