Uniqueness Analysis of Non-Unitary Matrix Joint Diagonalization

Abstract

Matrix Joint Diagonalization (MJD) is a powerful approach for solving the Blind Source Separation (BSS) problem. It relies on the construction of matrices which are diagonalized by the unknown demixing matrix. Their joint diagonalizer serves as a correct estimate of this demixing matrix only if it is uniquely determined. Thus, a critical question is under what conditions a joint diagonalizer is unique. In the present work we fully answer this question about the identifiability of MJD based BSS approaches and provide a general result on uniqueness conditions of matrix joint diagonalization. It unifies all existing results which exploit the concepts of non-circularity, non-stationarity, non-whiteness, and non-Gaussianity. As a corollary, we propose a solution for complex BSS, which can be formulated in a closed form in terms of an eigenvalue and a singular value decomposition of two matrices.

A Coordinate Descent Algorithm for Complex Joint Diagonalization Under Hermitian and Transpose Congruences

“…Looking at relations (23) and (27), the corresponding variables appear to be conjugated. By taking the conjugate of the second equation, the system we have to solve (32) can be simply written as (33) When is invertible, the two solutions of the above system can be analytically derived and are written as (34) where it is very well known that…”

“…Notice that it has a link with the uniqueness of the PARAFAC tensor decomposition [38]. In [32], the two complex cases were considered. Let us briefly recall the result.…”

“…In all cases, the matrix set should contain a sufficient number of matrices (greater than or equal to two) to ensure identifiability. This problem has been considered in [2], [9], [28]- [30] for source separation and in [31], [32] for joint matrix diagonalization. In such blind identification problem, it is well known that the searched diagonalizing matrix can be identified only up to the product by a permutation matrix and up to the product by an invertible diagonal matrix.…”

A Descent Algorithm-Based Parallel Variable Dual Matrix Model and Application to Blind Source Separation

“…By simultaneously avoiding D i , P j converging to singular matrices, A and W degenerate into nonfull-rank matrices; their penalty terms are added to the cost function. Inspired by Xi-Lin Li and Xian-Da Zhang, we incorporate penalty terms P( D i ) = −β log | D i | 2 , which are different to those in [8] for easier to compute their gradient functions, into cost function (7) to prevent square matrices D i degenerating into undesired solutions (singular matrices). Where β is a very small positive number, log is the abbreviation for log e , and |·| denotes the determinant of the squared matrix.…”

“…Obviously, this additional constraint may restrict general application areas of JD. As a consequence, quite a few algorithms, which remove the pre-whitening operation as well as relax the positive definiteness assumption on M, have been proposed in recent years [7,9,13,16,17]. However, matrix A sometimes converges a zero solution in the JD model; even under the amendment proposed by Li and Zhang [8], it hardly converges to a separable solution of the BSS problem even the cost function converges to zero, because these models do not consider the numerical relation between the separating matrix and the mixing matrix of BSS.…”