Bridge Simulation and Metric Estimation on Lie Groups

TL;DR

This paper introduces a simulation scheme for Brownian bridges on Lie groups, demonstrating measure absolute continuity and applying importance sampling to estimate metrics on SO(3).

Contribution

It generalizes Euclidean Brownian bridge simulation to Lie groups and develops a method for metric estimation using guided processes.

Findings

Simulation scheme for Brownian bridges on Lie groups

Absolute continuity of bridge measure with respect to guided process

Successful metric estimation on SO(3) using importance sampling

Abstract

We present a simulation scheme for simulating Brownian bridges on complete and connected Lie groups. We show how this simulation scheme leads to absolute continuity of the Brownian bridge measure with respect to the guided process measure. This result generalizes the Euclidean result of Delyon and Hu to Lie groups. We present numerical results of the guided process in the Lie group $\SO(3)$. In particular, we apply importance sampling to estimate the metric on $\SO(3)$ using an iterative maximum likelihood method.