A pseudo-Riemannian-gradient approach to the least-squares problem on the real symplectic group

The present paper discusses the problem of geodesic least-squares over the real symplectic group of matrices Sp(2n,∝). As the space Sp(2n,∝) is a non-compact Lie group, it is convenient to endow it with a pseudo-Riemannian geometry instead of a Riemannian one. Indeed, a pseudo-Riemannian metric allows the computation of geodesic arcs and geodesic distances in closed form.