Algorithms on the Stiefel manifold for joint diagonalisation

This paper presents a novel approach to the unitary joint diagonalisation problem. We approach the problem as an optimisation problem over the manifold of unitary matrices and obtain steepest descent and Newton algorithms. Both algorithms converge to the same solution as the well known Joint Approximate Diagonalisation of Eigen-matrices (JADE) algorithm. However, our Newton algorithm achieves a quadratic convergence rate almost immediately if suitably initialised, whereas JADE can (in some cases) take up to 30 sweeps before it starts converging to the solution at a quadratic rate. Simulations illustrate that our steepest descent algorithm can be used to find a suitable starting point for our Newton algorithm which then refines our estimate at a quadratic rate. This two stage algorithm requires less iterations overall than JADE, with the cost per iteration being the same as JADE. Our methods, therefore, offer a novel alternative to JADE.