On Consistency and Asymptotic Uniqueness in Quasi-Maximum Likelihood Blind Separation of Temporally-Diverse Sources

Abstract

In its basic, fully blind form, Independent Component Analysis (ICA) does not rely on a particular statistical model of the sources, but only on their mutual statistical independence, and therefore does not admit a Maximum Likelihood (ML) estimation framework. In semi-blind scenarios statistical models of the sources are available, enabling ML separation. Quasi-ML (QML) methods operate in the (more realistic) fully-blind scenarios, simply by presuming some hypothesized statistical models, thereby obtaining QML separation. When these models are (or are assumed to be) Gaussian with distinct temporal covariance matrices, the (quasi-)likelihood equations take the form of a “Sequentially Drilled Joint Congruence” (SeDJoCo) transformation problem. In this work we state some mild conditions on the sources' true and presumed covariance matrices, which guarantee consistency of the QML separation when the SeDJoCo solution is asymptotically unique. In addition, we derive a necessary “Mutual Diversity” condition on these matrices for the asymptotic uniqueness of the SeDJoCo solution. Finally, we demonstrate the consistency of QML in various simulation scenarios.