This paper studies the problem of the blind extraction of a subset of bounded component signals from the observations of a linear mixture. In the first part of this paper, we analyze the geometric assumptions of the observations that characterize the problem, and their implications on the mixing matrix and latent sources. In the second part, we solve the problem by adopting the principle of minimizing the risk, which refers to the encoding complexity of the observations in the worst admissible situation.
This principle provides an underlying justification of several bounded component analysis (BCA) criteria, including the minimum normalized volume criterion of the estimated sources or the maximum negentropy-likelihood criterion with a uniform reference model for the estimated sources. This unifying framework can explain the differences between the criteria in accordance with their considered hypotheses for the model of the observations. This paper is first presented for the case of the extraction of a complex and multidimensional source, and later is particularized for the case of the extraction of subsets of 1-D complex sources. The results also hold true in the case of real signals, where the obtained criteria for the extraction of a set of 1-D sources usually coincide with the existing BCA criteria.