Divergences on Symmetric Cones and Medians

We are concerned with divergences on the Cartan–Hadamard Riemannian manifold of symmetric cones, self-dual homogeneous cones in Euclidean spaces, and related optimization problems. We introduce a parameterized version of fidelity on symmetric cones, namely sandwiched quasi-relative entropies, and construct a one-parameter family of divergences based on these entropies. We consider the median minimization problem of finite points over these divergences and establish existence and uniqueness of minimizer. The global linear rate convergence of a gradient projection algorithm for solving the median minimization problem is analyzed based on the derived upper bound of the condition number of the Hessian function.