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 demixing matrix. The separation is based on the non-Gaussianity of the source of interest, while the remaining 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 to the extraction of a vector component when the extraction proceeds jointly from a set of instantaneous mixtures. Throughout this paper, we address the problem concerning the size of the region of convergence for which the algorithms guarantee the extraction of the desired source. We show that the size is influenced by the signal-to-interference ratio on the input channels. Simulations comparing several algorithms confirm this observation. They show a different size of the region of convergence under a scenario in which 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.