Transportless conjugate gradient for optimization on Stiefel manifold

Abstract

In this paper, we focus on building an optimization scheme over the Stiefel manifold that maintains each iterate feasible. We focus on conjugate gradient methods and compare our scheme to the Riemannian optimization approach. We parametrize the Stiefel manifold using the polar decomposition to build an optimization problem over a vector space, instead of a Riemannian manifold. The result is a conjugate gradient method that averts the use of a vector transport, needed in the Riemannian conjugate gradient method. The performance of our method is tested on a variety of numerical experiments and compared with those of three Riemannian optimization methods.