Newton-like methods for numerical optimization on manifolds

Many problems in signal processing require the numerical optimization of a cost function, which is defined on a smooth manifold. Especially, orthogonally or unitarily constrained optimization problems tend to occur in signal processing tasks involving subspaces. In this paper we consider Newton-like methods for solving these types of problems. Under the assumption that the parameterization of the manifold is linked to so-called Riemannian normal coordinates our algorithms can be considered as intrinsic Newton methods. Moreover, if there is not such a relationship, we still can prove local quadratic convergence to a critical point of the cost function by means of analysis on manifolds. Our approach is demonstrated by a detailed example, i.e., computing the dominant eigenspace of a real symmetric matrix.