Bridge Simulation and Metric Estimation on Lie Groups and Homogeneous Spaces

TL;DR

This paper develops methods for simulating Brownian bridges on Lie groups and homogeneous spaces, and uses these to estimate invariant Riemannian metrics and fit distributions to data on these spaces.

Contribution

It introduces novel simulation schemes for Brownian bridges on Lie groups and homogeneous spaces, and proposes an estimation method for invariant metrics and distributions from data.

Findings

Successful simulation of Brownian bridges on Lie groups and spaces

Effective estimation of invariant Riemannian metrics from samples

New parametric distribution families on spheres and positive tensors

Abstract

We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from samples. We subsequently show how pushing forward the distributions generated by Brownian motions on the group results in distributions on homogeneous spaces that exhibit non-trivial covariance structure. The pushforward measure gives rise to new parametric families of distributions on commonly occurring spaces such as spheres and symmetric positive tensors. We extend the estimation scheme to fit these distributions to homogeneous space-valued data. We demonstrate both the simulation schemes and estimation procedures on Lie groups and homogenous spaces, including $\SPD(3) = \GL_+(3)/\SO(3)$ and $\mathbb S^2 = \SO(3)/\SO(2)$.