Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices

Abstract. The present paper discusses the question of formulating and solving minimal-distance problems over the group-manifold of real symplectic matrices. In order to tackle the related optimization problem, the real symplectic group is regarded as a pseudo-Riemannian manifold and a metric is chosen that affords the computation of geodesic arcs in closed forms. Then, the considered minimal-distance problem can be solved numerically via a gradient steepest descent algorithm implemented through a geodesic-stepping method. The minimal-distance problem investigated in this paper relies on a suitable notion of distance – induced by the Frobenius norm – as opposed to the natural pseudo-distance that corresponds to the pseudo-Riemannian metric that the real symplectic group is endowed with. Numerical tests about the computation of the empirical mean of a collection of symplectic matrices illustrate the discussed framework.