Solving Weighted Orthogonal Procrustes Problems via a Projected Gradient Method

This paper proposes a family of line--search methods to deal with weighted orthogonal procrustes problems. In particular, the proposed family uses a search direction based on a convex combination between the Euclidean gradient and the Riemannian gradient of the cost function. The non--monotone line--search of Zhang and Hager, and an adaptive Barzilai--Borwein step--size are the chosen tools, in order to speed up the convergence of the new family of methods. One of the extremes of that convex combination is reduced to well--known spectral projected gradient method, while the another one can be interpreted as a Riemannian steepest descent method. To preserve feasibility of all the iterates, we use a projection operator based on the singular value decomposition, which can be computed efficiently via the spectral decomposition of an appropriate matrix. In addition, we prove that the entire uncountable collection of search directions satisfies a descent condition. Some numerical experiments are provided in order to demonstrate the effectiveness of the new approach.