Spherical Hamiltonian Monte Carlo
for Constrained Target Distributions
Shiwei Lan∗,
Bo Zhou∗ ,
Babak Shahbaba†
January 8, 2014
Supplementary: geodesic on a sphere
To find the geodesic on a sphere, we need to solve the following equations:
θ̇ = v
v̇ = −v T Γv
(1)
(2)
for which we need to calculate the Christoffel symbols, Γ, first. Note that the (i, j)-th element of GS is
4
2
2
.
+ 2θi θj θk /θD+1
, and the (i, j, k)-th element of dGS is gij,k = (δik θj + θi δjk )/θD+1
gij = δij + θi θj /θD+1
Therefore
1
Γkij = g kl [glj,i + gil,j − gij,l ]
2
2
= θk [δij + θi θj /θD+1
] = [GS ⊗ θ]ijk
Using these results, we can write equation (2) as v̇ = −v T GS vθ = −kṽk22 θ. Further, we have
p
θ̇D+1 = dtd 1 − kθk22 = vD+1
v̇D+1 =
T
= −kṽk22 θD+1
v
− dtd θθD+1
Therefore, we can rewrite the geodesic equations (1)(2) as
θ̃˙ = ṽ
ṽ˙ = −kṽk22 θ̃
Multiplying both sides of Equation (4) by ṽ T to obtain
differential equations as follows:
θ̃(t) = θ̃(0) cos(kṽ(0)k2 t) +
dkṽk22
dt
(3)
(4)
= 0, we can solve the above system of
ṽ(0)
sin(kṽ(0)k2 t)
kṽ(0)k2
ṽ(t) = −θ̃(0)kṽ(0)k2 sin(kṽ(0)k2 t) + ṽ(0) cos(kṽ(0)k2 t)
∗
†
Department of Statistics, University of California, Irvine, USA.
Department of Statistics and Department of Computer Science, University of California, Irvine, USA.
1