Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices
Simone Fiori · SIAM Journal on Matrix Analysis and Applications · 2011
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 pseudodistance 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.