On minimizing a quadratic function on Stiefel manifold

Abstract

In this paper we propose a novel approach to a particular quadratic programming problem, when the optimization is performed over the set O(3,2) of 3×2 Stiefel matrices. We rewrite the original nonconvex problem as a semi-definite programming problem, by computing a convex hull (tight convex relaxation) of a certain set of matrices. We give an efficient, quick algorithm for the minimization of a quadratic function over Stiefel manifold. We report some numerical experiments to illustrate the tightness of the convex approximation obtained by the two aforementioned methods ("standard" and ours). Our result is of immediate interest in Computer Vision, including Structure-from-Motion (SfM) problems, and 2D-3D registration.