Iterative methods for the canonical decomposition of multi-way arrays: Application to blind underdetermined mixture identification

Two main drawbacks can be stated in the Alternating Least Square (ALS) algorithm used to fit the CANonical Decomposition (CAND) of multi-way arrays. First its slow convergence caused by the presence of collinearity between factors in the multi-way array it decomposes. Second its blindness to Hermitian symmetries of the considered arrays. Enhanced Line Search (ELS) scheme was found to be a good way to cope with the slow convergence of the ALS algorithm together with a partial use of the Hermitian symmetry. However, to our knowledge, required equations to perform the latter scheme are only given in the case of third and fifth order arrays. Therefore, our first contribution consists in generalizing the ELS procedure to the case of complex arrays of any order greater than three. Our second contribution is another improvement of the ALS scheme, able to profit from Hermitianity and positive semi-definiteness of the considered arrays. It consists in resorting to the CAND of a third order array having one unitary loading matrix. An iterative algorithm is then proposed alternating between Procrustes problem solving and the computation of rank-one matrix approximations.