Efficient Line Search Methods for Riemannian Optimization Under Unitary Matrix Constraint

This paper proposes a Riemannian geometry approach for optimization under unitary matrix constraint. We introduce two novel line search methods which are used together with a steepest descent algorithm on the Lie group of unitary matrices U(n). The proposed approach fully exploits special properties that only appear on U(n), and do not appear on the Euclidean space or arbitrary Riemannian manifolds. These properties induce an almost periodic behavior of the cost function along geodesies. Consequently, the resulting step size selection rule outperforms efficient methods such as Armijo rule [1] in terms of complexity. We test the proposed optimization algorithm in a blind source separation application for MIMO systems by using the joint diagonalization approach [2]. The proposed algorithm converges faster than the classical JADE algorithm [2].