Gradient Algorithms for Complex Non-Gaussian Independent Component/Vector Extraction, Question of Convergence

We revise the problem of extracting one independent component from an instantaneous linear mixture of signals. The mixing matrix is parameterized by two vectors, one column of the mixing matrix and one row of the de-mixing matrix. The separation is based on the non-Gaussianity of the source of interest, while the other background signals are assumed to be Gaussian. Three gradient-based estimation algorithms are derived using the maximum likelihood principle and are compared with the Natural Gradient algorithm for Independent Component Analysis and with One-unit FastICA based on negentropy maximization. The ideas and algorithms are also generalized for the extraction of a vector component when the extraction proceeds jointly from a set of instantaneous mixtures. Throughout the paper, we address the problem of the size of the region of convergence for which the algorithms guarantee the extraction of the desired source. We show how that size is influenced by the ratio of powers of the sources within the mixture. Simulations confirm this observation where several algorithms are compared. They show various convergence behavior in a scenario where the source of interest is dominant or weak. Here, our proposed modifications of the gradient methods taking into account the dominance/weakness of the source show improved global convergence property.