Bridge Simulation on Lie Groups and Homogeneous Spaces with Application to Parameter Estimation

We present three simulation schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces and use numerical results of the guided processes in the Lie group $\SO(3)$ and on the homogeneous spaces $\mathrm{SPD}(3) = \mathrm{GL}_+(3)/\mathrm{SO}(3)$ and $\mathbb S^2 = \mathrm{SO}(3)/\mathrm{SO}(2)$ to evaluate our sampling scheme. Brownian motions on Lie groups can be defined via the Laplace-Beltrami of a left- (or right-)invariant Riemannian metric. Given i.i.d. Lie group-valued samples on $\mathrm{SO}(3)$ drawn from a Brownian motion with unknown Riemannian metric structure, the underlying Riemannian metric on $\mathrm{SO}(3)$ is estimated using an iterative maximum likelihood (MLE) method. Furthermore, the re-sampling technique is applied to yield estimates of the heat kernel on the two-sphere considered as a homogeneous space. Comparing this estimate to the truncated version of the closed-form expression for the heat kernel on $\mathbb S^2$ serves as a proof of concept for the validity of the sampling scheme on homogeneous spaces.