An intrinsic Cram\'er-Rao bound on Lie groups

TL;DR

This paper derives an intrinsic Cramér-Rao bound specifically for parameters on Lie groups, simplifying the derivation by leveraging the structure of Lie groups and comparing it to previous Riemannian manifold results.

Contribution

It provides a new, simplified derivation of the intrinsic Cramér-Rao bound for Lie groups using invariant metrics, connecting it to prior Riemannian manifold bounds.

Findings

Bound is simplified using Lie group structure

Results coincide with previous bounds for bi-invariant metrics

Provides a natural derivation for Lie group parameters

Abstract

In his 2005 paper, S.T. Smith proposed an intrinsic Cram\'er-Rao bound on the variance of estimators of a parameter defined on a Riemannian manifold. In the present technical note, we consider the special case where the parameter lives in a Lie group. In this case, by choosing, e.g., the right invariant metric, parallel transport becomes very simple, which allows a more straightforward and natural derivation of the bound in terms of Lie bracket, albeit for a slightly different definition of the estimation error. For bi-invariant metrics, the Lie group exponential map we use to define the estimation error, and the Riemannian exponential map used by S.T. Smith coincide, and we prove in this case that both results are identical indeed.