  The joint approximate diagonalization of non-commuting symmetric matrices is
an important process in independent component analysis. This problem can be
formulated as an optimization problem on the Stiefel manifold that can be
solved using Riemannian optimization techniques. Among the available
optimization techniques, this study utilizes the Riemannian Newton's method for
the joint diagonalization problem on the Stiefel manifold, which has quadratic
convergence. In particular, the resultant Newton's equation can be effectively
solved by means of the Kronecker product and the vec and veck operators, which
reduce the dimension of the equation to that of the Stiefel manifold. Numerical
experiments are performed to show that the proposed method improves the
accuracy of the approximate solution to this problem. The proposed method is
also applied to independent component analysis for the image separation
problem. The proposed Newton method further leads to a novel and fast
Riemannian trust-region Newton method for the joint diagonalization problem.
