Quaternion ICA From Second-Order Statistics

This paper addresses the independent component analysis (ICA) of quaternion random vectors. In particular, we focus on the Gaussian case and therefore only consider the quaternion second-order statistics (SOS), which are given by the covariance matrix and three complementary covariance matrices. First, we derive the necessary and sufficient conditions for the identifiability of the quaternion ICA model, which are based on the definition of the properness profile of a quaternion random variable and more specifically on the concept of rotationally equivalent properness profiles. Second, we show that the maximum-likelihood (ML) approach to the quaternion ICA problem reduces to the approximated joint diagonalization (AJD) of the sample-mean estimates of the covariance and complementary covariance matrices. Unlike the complex case, these four matrices cannot be simultaneously diagonalized in general, and we have to resort to a particular AJD algorithm. The proposed technique, which can be seen as a quasi-Newton method, is based on the local approximation of the nonconvex ML-ICA cost function (a measure of the entropy loss due to the residual correlation among the estimated quaternion sources), and it provides a satisfactory solution of the quaternion ICA model. The performance of the proposed quaternion ML-ICA algorithm, as well as its relationship to the identifiability conditions, are illustrated by means of several numerical examples.