Maximum likelihood ICA of quaternion Gaussian vectors

This work considers the independent component analysis (ICA) of quaternion random vectors. In particular, we focus on the Gaussian case, and therefore the ICA problem is solved by exclusively exploiting the second-order statistics (SOS) of the observations. In the quaternion case, the SOS of a random vector are given by the covariance matrix and three complementary covariance matrices. Thus, quaternion ICA amounts to jointly diagonalizing these four matrices. Following a maximum likelihood (ML) approach, we show that the ML-ICA problem reduces to the minimization of a cost function, which can be interpreted as a measure of the entropy loss due to the correlation among the estimated sources. In order to solve the non-convex ML-ICA problem, we propose a practical quasi-Newton algorithm based on quadratic local approximations of the cost function. Finally, the practical performance and potential application of the proposed technique is illustrated by means of numerical examples.