Means of Random Variables in Lie Groups

TL;DR

This paper reviews various definitions of mean and covariance for random variables on matrix Lie groups, analyzing their relationships and conditions for optimality to guide practical applications.

Contribution

It provides a comprehensive comparison of mean definitions on Lie groups, clarifies their relationships, and offers guidance for selecting appropriate means in applications.

Findings

Different mean definitions have distinct properties and conditions for optimality.

Group-theoretic means minimize specific least-squares cost functions under certain conditions.

The choice of inner product on the Lie algebra influences the mean definitions.

Abstract

The concepts of mean (i.e., average) and covariance of a random variable are fundamental in statistics, and are used to solve real-world problems such as those that arise in robotics, computer vision, and medical imaging. On matrix Lie groups, multiple competing definitions of the mean arise, including the Euclidean, projected, distance-based (i.e., Fr\'echet and Karcher), group-theoretic, and parametric means. This article provides a comprehensive review of these definitions, investigates their relationships to each other, and determines the conditions under which the group-theoretic means minimize a least-squares type cost function. We also highlight the dependence of these definitions on the choice of inner product on the Lie algebra. The goal of this article is to guide practitioners in selecting an appropriate notion of the mean in applications involving matrix Lie groups.