Algorithm comparison for Karcher mean computation of rotation matrices and diffusion tensors

This paper concerns the computation, by means of gradient and Newton methods, of the Karcher mean of a finite collection of points, both on the manifold of 3×3 rotation matrices endowed with its usual bi-invariant metric and on the manifold of 3×3 symmetric positive definite matrices endowed with its usual affine invariant metric. An explicit expression for the Hessian of the Riemannian squared distance function of these manifolds is given. From this, a condition on the step size of a constant step gradient method that depends on the data distribution is derived. These explicit expressions make a more efficient implementation of the Newton method possible and it is shown that the Newton method outperforms the gradient method in some cases.