A Riemannian-steepest-descent approach for optimization on the real symplectic group

In this paper, we first give the geodesic in closed form on the real symplectic group endowed with a Riemannian metric and then study a geodesic‐based Riemannian‐steepest‐descent approach to compute the empirical average out of a set of symplectic matrices. The devised averaging algorithm is compared with the Euclidean gradient algorithm and the extended Hamiltonian algorithm. Simulation examples show that the convergence of the geodesic‐based Riemannian‐steepest‐descent algorithm is the fastest among the 3 considered algorithms.