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

Abstract

This paper deals with the problem of joint complex matrix diagonalization by considering both the Hermitian and transpose congruences. We address the general case where the searched diagonalizing matrix is a priori nonunitary. Based on the minimization of a quadratic criterion, we propose a flexible algorithm in the sense that it allows to directly consider a rectangular diagonalizing matrix and to take into consideration both the Hermitian and transpose congruences within the same framework. The proposed algorithm is a coordinate descent algorithm that is based on an approximate criterion leading to the analytical derivation of the minima arguments. Computer simulations are drawn to illustrate the usefulness and performances of the algorithm and a comparison to state-of-the-art algorithms is presented. Finally, an application to independent component analysis based on fourth-order statistics is also presented.

Hybrid Joint Diagonalization Algorithms

“…We compare our proposed algorithms with the following ones: Second-Order Blind Identification (SOBI) [1], Fast Approximate Joint Diagonalization (FAJD) [20], Hybrid FAJD (H-FAJD) [7], NOODLES [8], and H-NOODLES [8] 6 . Par-ticularly, we simulate cases representing adverse conditions under which JD can be difficult.…”

“…As expected, and based on our simulation experiments, the extended algorithms provide improved HJD performance as compared to state-of-the-art methods. 8 Note that for ARO-HJD, the performance index is applied to matrix P = V T A.…”

“…It is mostly encountered when dealing with the statistics of multivariate non-circular complex data (see for example [6]). Among the existing solutions for this HJD problem one can cite the extended version of FAJD proposed in [7], the NOODLES algorithm proposed in [8], which relies on a natural-gradient technique, the algorithm proposed in [9] based on weighted least-squares (WLS) criterion, and the alternating least squares (ALS) algorithm proposed in [10], which considers A H and A T as different variables during the iterative process. All of these methods consider the non-orthogonal case where the matrix A is non-unitary.…”

Preconditioned optimization algorithms solving the problem of the non unitary joint block diagonalization: application to blind separation of convolutive mixtures

“…Recently, a more general problem has been addressed: that of the joint decomposition of a combination of given sets of complex matrices that can follow potentially different decompositions (for example when noncircular complex valued signals are considered and when people want to exploit additional statistical information by using both Hermitian and complex symmetric matrices simultaneously) (Moreau and Adali 2013;Trainini and Moreau 2014;Zeng et al 2009). Such issues can arise in various signal processing problems, among which are blind identification, separation or multidimensional deconvolution.…”

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

“…Nowadays, attention has focused on the non-orthogonal joint diagonalization of matrices, see e.g. [7], [8] and references there in. A non-orthogonal diagonalization is important mainly because it allows to skip a first processing step (called whitening in source separation) that limits the performances in practice.…”