Independent Component Analysis Based on Learning Updating with Forms of Matrix Transformations and the Diagonalization Principle

Power iteration (PI) is a new algorithm for independent component analysis (ICA), which has some desired features. One of features is that the algorithm does not include any predetermined parameter such as the learning step size as in the gradient-based algorithm, which is especially expected for ICA applications with unknown types of sources. Another feature is that, in each iteration, the updating of ICA matrix is fully-multiplicative, rather than the additive updating in the conventional gradient learning algorithms. The criterion for the independence between outputs is based on diagonality of a non-linearized covariance matrix that is defined by ICA outputs and non-linear mapped ICA outputs. In this paper, we study the algorithm mathematically to analyze why and how the algorithm works. We show that in the algorithm the learning updating is in the form of matrix transformation acting on the ICA matrix. Such a updating allows a finite scale learning, which is essentially different from that only the small enough scale learning controlled by the learning step size is allowed, in a gradient-based algorithm. We also analyze the relation with the well known algorithms, such as, the Bussgang algorithm and the non-linear PCA