Parameter Estimation on Homogeneous Spaces

TL;DR

This paper develops a group-theoretic approach to parameter estimation on homogeneous spaces, simplifying the Fisher Information Metric and Cramér-Rao Bound analysis by leveraging symmetry structures, with applications in robotics and sensor networks.

Contribution

It introduces a novel group-theoretic framework for Fisher information and CRB on homogeneous spaces, extending classical Riemannian methods with practical engineering applications.

Findings

Homogeneous spaces facilitate easier analysis of Fisher information.

The group-theoretic CRB provides tighter bounds in symmetric models.

Applications demonstrate the framework's effectiveness in robotics and sensor localization.

Abstract

The Fisher Information Metric (FIM) and the associated Cram\'er-Rao Bound (CRB) are fundamental tools in statistical signal processing, which inform the efficient design of experiments and algorithms for estimating the underlying parameters. In this article, we investigate these concepts for the case where the parameters lie on a homogeneous space. Unlike the existing Fisher-Rao theory for general Riemannian manifolds, our focus is to leverage the group-theoretic structure of homogeneous spaces, which is often much easier to work with than their Riemannian structure. The FIM is characterized by identifying the homogeneous space with a coset space, the group-theoretic CRB and its corollaries are presented, and its relationship to the Riemannian CRB is clarified. The application of our theory is illustrated using two examples from engineering: (i) estimation of the pose of a robot and (ii) sensor network localization. In particular, these examples demonstrate that homogeneous spaces provide a natural framework for studying statistical models that are invariant with respect to a group of symmetries.