Fast Jacobi algorithm for non-orthogonal joint diagonalization of non-symmetric third-order tensors

We consider the problem of non-orthogonal joint diagonalization of a set of non-symmetric real-valued third-order tensors. This appears in many signal processing problems and it is instrumental in source separation. We propose a new Jacobi-like algorithm based on an LU decomposition of the so-called diagonalizing matrices. The parameters estimation is done entirely analytically following a strategy based on a classical inverse criterion and a fully decoupled estimation. One important point is that the diagonalization is directly done on the set of third-order tensors and not on their unfolded version. Computer simulations illustrate the overall good performances of the proposed algorithm.