Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem

Higher-order multivariate statistics are addressed. A theoretical discussion is followed by an original application. A special (index-free) tensor formalism to express fourth-order multivariate statistics is proposed. A quadricovariance tensor which contains the fourth-order joint cumulants is defined. In this formalism, the quadricovariance tensor and its eigen-matrices are natural fourth-order generalizations of second-order covariance and eigenvectors, allowing direct extension of many standard second-order method to fourth-order. The idea of an eigen-matrix is then proposed as a solution to the blind source separation problem. The task is to separate a mixture of N independent non-Gaussian signals received on an array of sensors when no information is available about propagation conditions or array geometry (blind situation). This can be achieved only by resorting to higher-order information. Quadricovariance eigen-matrices give a direct solution to this problem