A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups

Motivated by applications in fuzzy control, robotics and vision, this paper considers the problem of computing the centre of mass (precisely, the Karcher mean) of a set of points defined on a compact Lie group, such as the special orthogonal group consisting of all orthogonal matrices with unit determinant. An iterative algorithm, whose derivation is based on the geometry of the problem, is proposed. It is proved to be globally convergent. Interestingly, the proof starts by showing the algorithm is actually a Riemannian gradient descent algorithm with fixed step size.