On the convergence of ICA algorithms with weighted orthogonal constraint

• The symmetric Minimum Distance weighted Unitary Mapping of nonsingular square matrix is obtained.
• A characterization of the stationary points corresponding to weighted orthogonal constrained ICA algorithms using symmetric Minimum Distance weighted Unitary Mapping is obtained.
• The monotonic convergence property of the weighted orthogonal constrained fixed point ICA algorithms using symmetric Minimum Distance weighted Unitary Mapping for convex contrast function is proved, which is further extended to nonconvex contrast function case.

A kind of weighted orthogonal constrained independent component analysis (ICA) algorithms with weighted orthogonalization for achieving this constraint is proposed recently. It has been proved in the literature that weighted orthogonal constrained ICA algorithms keep the equivariance property and have much better convergence speed, separation ability and steady state misadjustment, but the convergence is not yet analyzed in the published literature. The goal of this paper is to fill this gap. Firstly, a characterization of the stationary points corresponding to these algorithms using symmetric Minimum Distance Weighted Unitary Mapping (MDWUM) for achieving the weighted orthogonalization is obtained. Secondly, the monotonic convergence of the weighted orthogonal constrained fixed point ICA algorithms using symmetric MDWUM for convex contrast function is proved, which is further extended to nonconvex contrast functions case by adding a weighted orthogonal constraint term onto the contrast function. Together with the boundedness of contrast function, the convergence of fixed point ICA algorithms with weighted orthogonal constraint using symmetric MDWUM is implied. Simulation experiments results show that the adaptive ICA algorithms using symmetric MDWUM are better in terms of accuracy than the ones with pre-whitening, and the fixed-point ICA algorithms using symmetric MDWUM converge monotonically.