Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation

Approximate joint diagonalization of a set of matrices is an essential tool in many blind source separation (BSS) algorithms. A common measure of the attained diagonalization of the set is the weighted least-squares (WLS) criterion. However, most well-known algorithms are restricted to finding an orthogonal diagonalizing matrix, relying on a whitening phase for the nonorthogonal factor. Often, such an approach implies unbalanced weighting, which can result in degraded performance. In this paper, we propose an iterative alternating-directions algorithm for minimizing the WLS criterion with respect to a general (not necessarily orthogonal) diagonalizing matrix. Under some mild assumptions, we prove weak convergence in the sense that the norm of parameters update is guaranteed to fall below any arbitrarily small threshold within a finite number of iterations. We distinguish between Hermitian and symmetrical problems. Using BSS simulations results, we demonstrate the improvement in estimating the mixing matrix, resulting from the relaxation of the orthogonality restriction.