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F. Nielsen
Fig. 7.6 Dual parallel transport of tangent vector ˙
γ ∗
pq (0) at T p along the ∇ ∗ -geodesic: Since the
dual geodesic γ ∗
pq (t) is a ∇ ∗ -autoparallel curve, the tangent vector at ˙
γ ∗
pq (0) is transported into
tangent vectors along that dual geodesic
= u
i
v i g p
e i | p , e
∗i
p
, (7.32)
= g p (u, v).
(7.33)
Figure 7.6 displays the parallel transport of the tangent vector ˙
γ
∗
pq (0) from T p
to T γ ∗
pq (t) for several time steps t. Although the dual parallel transport preserves the
metric, the length of a vector transported by either the primal or the dual parallel
transport varies.
Historically, the dual parallel transport coupled to the metric tensor was studied
independently by Norden [25] and Sen [28] in the 1930s and 1940s, respectively.
7.2.3 Dual Pythagorean Theorems
The divergence D F ( p : q) from a point p to a point q can be expressed using dual
Bregman divergences as follows:
D F ( p : q) = B F (θ ( p) : θ(q)) = D
∗
(q : p) = B F ∗ (η(q) : η( p)),
(7.34)
where D
∗
F ( p : q) := D F (q : p) denotes the reverse divergence, and
B F (θ 1 : θ 2 ) = F(θ 1 ) − F(θ 2 ) − (θ 1 − θ 2 )
∇ F(θ 2 ),
(7.35)
is the Bregman divergence associated to generator F. Bregman divergences generalize the squared Euclidean distance with the relative entropy, and thus offer a nice
framework to unify or develop generic algorithms [4]. Bregman divergences proved
useful in machine learning [5] and topological data analysis [7] among others.
Now, consider two smooth curves c 1 (t) and c 2 (t) intersecting at a point p =
c 1 (0) = c 2 (0). Denote by ˙
c 1 (0) =
d
dt
c 1 (t)
t=0
and ˙
c 2 (0) =
d
dt
c 2 (t)
t=0
the tangent
vectors to the curves at point p, belonging to the tangent plane T p . Curve c 1 is said
orthogonal to curve c 2 at p, i.e., c 1 (t) ⊥ p c 2 (t), iff. g p ( ˙
c 1 (0), ˙
c 2 (0)) = 0.