Newton Method for Riemannian Centroid Computation in Naturally Reductive Homogeneous Spaces

We address the problem of computing the Riemannian centroid of a constellation of points in a naturally reductive homogeneous manifold. We note that many interesting manifolds used in engineering (such as the special orthogonal group, Grassman, sphere, positive definite matrices) possess this structure. We develop an intrinsic Newton scheme for the centroid computation. This is achieved by exploiting a formula that we introduce for obtaining the Hessian of the squared Riemannian distance on naturally reductive homogeneous spaces. Some results of finding the centroid of a constellation of points in these spaces are presented, which evidence the quadratic convergence of the Newton method derived herein. These computer simulation results show that, as expected, the Newton method has a faster convergence rate than the usual gradient-based approaches