Stability analysis of complex maximum likelihood ica using Wirtinger calculus

The desirable asymptotic optimality properties of the maximum likelihood (ML) estimator make it an attractive solution for performing independent component analysis (ICA) as well. Wirtinger calculus is shown to provide an attractive framework for the derivation and analysis of complex-valued algorithms using nonlinear functions, and hence of ICA algorithms as well. Local stability analysis of complex ICA based on ML presents a unique challenge, since in addition to the need for computation of derivatives, the Hessian of a matrix quantity needs to be evaluated, and for the complex case, it assumes a significantly more complicated form than the real-valued case. In this paper, we demonstrate how Wirtinger calculus allows the use of an elegant approach proposed by Amari et al. (1997) in the analysis, thus enabling the derivation of the conditions for local stability of complex ML ICA. We further study the implications of the conditions for a generalized Gaussian density model.