Joint computation of principal and minor components using gradient dynamical systems over stiefel manifolds

Abstract

This paper presents several dynamical systems for simultaneous computation of principal and minor subspaces of a symmetric matrix. The proposed methods are derived from optimizing cost functions which are chosen to have optimal values at vectors that are linear combinations of extreme eigenvectors of a given matrix. Necessary optimality conditions are given in terms of a gradient of certain cost functions over a Stiefel manifold. Stability analysis of equilibrium points of six algorithms is established using Liapunov direct method.