Nonholonomic Orthogonal Learning Algorithms for Blind Source Separation

Independent component analysis or blind source separation extracts independent signals from their linear mixtures without assuming prior knowledge of their mixing coefficients. It is known that the independent signals in the observed mixtures can be successfully extracted except for their order and scales. In order to resolve the indeterminacy of scales, most learning algorithms impose some constraints on the magnitudes of the recovered signals. However, when the source signals are nonstationary and their average magnitudes change rapidly, the constraints force a rapid change in the magnitude of the separating matrix. This is the case with most applications (e.g., speech sounds, electroencephalogram signals). It is known that this causes numerical instability in some cases. In order to resolve this difficulty, this article introduces new nonholonomic constraints in the learning algorithm. This is motivated by the geometrical consideration that the directions of change in the separating matrix should be orthogonal to the equivalence class of separating matrices due to the scaling indeterminacy. These constraints are proved to be nonholonomic, so that the proposed algorithm is able to adapt to rapid or intermittent changes in the magnitudes of the source signals. The proposed algorithm works well even when the number of the sources is overestimated, whereas the existent algorithms do not (assuming the sensor noise is negligibly small), because they amplify the null components not included in the sources. Computer simulations confirm this desirable property.