On the convergence of ICA algorithms with symmetric orthogonalization

Abstract

Independent component analysis (ICA) problem is often posed as the maximization/minimization of an objective/cost function under a unitary constraint, which presumes the prewhitening of the observed mixtures. The parallel adaptive algorithms corresponding to this optimization setting, where all the separators are jointly trained, are typically implemented by a gradient-based update of the separation matrix followed by the so-called symmetrical orthogonalization procedure to impose the unitary constraint. This article addresses the convergence analysis of such algorithms, which has been considered as a difficult task due to the complication caused by the minimum-(Frobenius or induced 2-norm) distance mapping step. We first provide a general characterization of the stationary points corresponding to these algorithms. Furthermore, we show that fixed point algorithms employing symmetrical orthogonalization are monotonically convergent for convex objective functions. We later generalize this convergence result for nonconvex objective functions. At the last part of the article, we concentrate on the kurtosis objective function as a special case. We provide a new set of critical points based on Householder reflection and we also provide the analysis for the minima/maxima/saddle-point classification of these critical points.