A generalization of joint-diagonalization criteria for source separation

In the field of blind source separation, joint-diagonalization-based approaches constitute an important framework, leading to useful algorithms such as the popular joint approximate diagonalization of eigenmatrices (JADE) and simultaneous third-order tensor diagonalization (STOTD) algorithms. However, they are often restricted to the case of cumulants of order four. In this paper, we extend the results leading to JADE and STOTD to cumulants of any order greater than or equal to three by exhibiting a new family of contrast functions that constitutes then a unified framework for the above known results. This also leads us to generalize some links between contrast functions and joint-diagonalization criteria on which these algorithms are based. In turn, one contrast of the new family allows us to show that a function previously proposed as a separation criterion is also a contrast. Moreover, for the two generalized JADE and STOTD contrasts, the analytical optimal solution in the case of two sources is derived and shown to keep the same simple expression, whatever the cumulant order. Finally, some computer simulations illustrate the potential advantage one can take by considering statistics of different orders for the joint-diagonalization of cumulant matrices.