Hessian of the Riemannian distance function on connected locally symmetric spaces: centroid computation with a Newton method
The seminar addresses the problem of computing the Riemannian centroid of a constellation of points in a locally symmetric connected manifold, particularly ones with a naturally reductive homogeneous space structure. Note that many interesting manifolds used in engineering (such as the special orthogonal group, Grassman, sphere, positive definite matrices) possess this structure. An intrinsic Newton scheme for the centroid computation is thus made available. Some results of finding the centroid of a constellation of points in these spaces are presented, which evidences 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.
