An Algorithm to Compute Averages on Matrix Lie Groups

Averaging is a common way to alleviate errors and random fluctuations in measurements and to smooth out data. Averaging also provides a way to merge structured data in a smooth manner. The present paper describes an algorithm to compute averages on matrix Lie groups. In particular, we discuss the case of averaging over the special orthogonal group of matrices, the unitary group of matrices and the group of symmetric positive-definite matrices.

I. Introduction

Averaging is a common way to alleviate measurement errors, to cope with statistical fluctuations and to smooth out data. Computing the mean value or the median value of a set of scalar, real-valued measures is an easy task. In some circumstances, however, measures to process appear as structured data, like vectors or matrices, whose entries should further satisfy nonlinear constraints. We refer to such structured/constrained data by the generic term of “patterns.”