
140 Current Trends in Bayesian Methodology with Applications
and with the initial velocity v. Subroutines 4 and 5 make use of the fact that
Q
0
⊂ S
∞
, the unit hypersphere in L
2
space, which has well-known analytical
formulas for the exponentia l and inverse exponential mappings.
Subroutine 4 – Exponential Map: Given [q] ∈ S, v ∈ T
[q]
(S), an integer
n, and ǫ > 0,
1. If kvk < ǫ, return [p] = [q], else
2. Let δ = 1/n. For i = 1, ..., n
(a) Compute q
adv
= exp
q
(δv) on S
∞
via the formula q
adv
= cos(δkvk)q +
sin(δkvk)
v
kvk
.
(b) Project [q
adv
] to S us ing S ubroutine 1.
(c) Parallel translate v from T
[q]
(S) to T
[q
adv
]
(S) using Subroutine 3. Let
q = q
adv
.
3. Return [p