Eigen-structure of the Fourth-order Cumulant Tensor with Application to the Blind Source Separation Problem

This communication deals with higher-order multivariate statistics. A first theoretical part is followed by an original application. In the first part we propose a special (index-free) tensor formalism to express fourth-order multivariate statistics. A quadricovariance "tensor" is defined which contains the 4th-order joint cumulants. In our formalism, the quadricovariance tensor and its "eigen-matrices" are natural 4th-order generalisations of 2nd-order covariance and eigenvectors, allowing direct extension of many standard 2nd-order methods to 4th-order. The idea of eigen-matrix is then proposed as a solution to the "blind source separation problem". The task is to separate a mixture of N independent nonGaussian signals received on a 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 eigenmatrices give a direct solution to this problem.