Using Generic Order Moments for separation of dependent sources with linear conditional expectations

In this work, we approach the blind separation of dependent sources based only on a set of their linear mixtures. We prove that, when the sources have a pairwise dependence characterized by the linear conditional expectation (LCE) law, i.e. E[S<sub>i</sub>|S<sub>j</sub>] =ρ<sub>ij</sub>S<sub>j</sub> for i ≠ j, with ρ<sub>ij</sub> = E[S<sub>i</sub>S<sub>j</sub>] (correlation coefficient), we are able to separate them by maximizing or minimizing a Generic Order Moment (GOM) of their mixture defined by μ<sub>p</sub> = E[|α<sub>1</sub>S<sub>1</sub> + α<sub>2</sub>S<sub>2</sub>|<sup>p</sup>]. This general measure includes the higher order as well as the fractional moment cases. Our results, not only confirm some of the existing results for the independent sources case but also they allow us to explore new objective functions for Dependent Component Analysis. A set of examples illustrating the consequences of our theory is presented. Also, a comparison of our GOM based algorithm, the classical FASTICA and a very recently proposed algorithm for dependent sources, the Bounded Component Analysis (BCA) algorithm, is shown.

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