Algorithms for Complex ML ICA and Their Stability Analysis Using Wirtinger Calculus

Abstract

We derive a class of algorithms for independent component analysis (ICA) based on maximum likelihood (ML) estimation and perform stability analysis of natural gradient ML ICA with and without the constraint for unitary demixing matrix. In the process, we demonstrate how Wirtinger calculus facilitates derivations, and most importantly, performing second-order analysis in the complex domain and eliminates the need for making simplifying assumptions. We derive natural gradient complex ML ICA update rule and its variant with a unitary constraint, as well as a Newton algorithm for better convergence behavior. The conditions for local stability are derived and studied using a generalized Gaussian density (GGD) source model. When the sources are circular and non-Gaussian, we show analytically that both the ML and ML-unitary ICA update rules converge to the inverse of mixing matrix subject to a phase shift. When the sources are noncircular and non-Gaussian, we show that the nonunitary ML ICA update rule is more stable than the ML-unitary ICA update rule. When the sources are noncircular Gaussians, both update rules are stable only when the sources have distinct noncircularity indices. Simulation results are given to support these results.