A Fast Algorithm for Nonunitary Joint Diagonalization and Its Application to Blind Source Separation

A New Algorithm for Complex Non-Orthogonal Joint Diagonalization Based on Shear and Givens Rotations

“…Xu, Feng and Zheng generalized the latter to the complex NOJD in [19]. It minimizes CILS criterion and estimates the diagonalizing matrix V in an iterative scheme using the following form…”

“…The rotation parameters are now optimized in such a way we minimize the simplified JD criterion given in (19) for the transformed matrices:…”

“…Many other algorithms have been developed by considering specific criteria or constraints in order to avoid trivial and degenerate solutions [14] [15]. These algorithms can be listed as follow: QDiag [16] (Quadratic Diagonalization algorithm) developed by Vollgraf and Obermayer where the JD criterion is rearranged as a quadratic cost function; FAJD [17] (Fast Approximative Joint Diagonalization) developed by Li and Zhang where the diagonalizing matrix is estimated column by column; UWEDGE [14] (UnWeighted Exhaustive joint Diagonalization with Gauss itErations) developed by Tichavsky and Yeredor where numerical optimization is used to get the JD solution; JUST [18] (Joint Unitary and Shear Transformations) developed by Iferroudjene, Abed-Meraim and Belouchrani where the algebraic joint diagonalization is considered; CVFFDiag [19] [20] (Complex Valued Fast Frobenius Diagonalization) developed by Xu, Feng and Zheng where first order of Taylor expansion is used to minimize the JD criterion; ALS [21] (Alternating Least Squares) developed by Trainini and Moreau where the mixing and diagonal matrices are estimated alternatively by using least squares criterion and LUCJD [22] [23] (LU decomposition for Complex Joint Diagonalization) developed by Wang, Gong and Lin where the diagonalizing matrix is estimated by LU decomposition.…”

Nonnegative Joint Diagonalization by Congruence Based on LU Matrix Factorization

“…Therefore we propose to impose a nonnegativity constraint on the matrix in the JDC problem. Generally, can be estimated either indirectly or directly [4]: 1) Indirect algorithms, such as JAD [3], FFDIAG [22], QDIAG [18], LUJ1D [1], FLEXJD [21], J-DI [14] and CVFFDIAG [19], estimate from the inverse of a transformation matrix , which minimizes the non-diagonal parts of using the following criterion:…”

Comparative performance analysis of non orthogonal joint diagonalization algorithms

“…JD algorithms can be classified in orthogonal JD (OJD) methods which proceed in two steps: whitening and orthogonal diagonalization steps, and non orthogonal JD methods which avoid the whitening step and help improve the separation performance when applied in BSS [3]. In this paper, we provide a comparative performance analysis of major iterative CNOJD algorithms, namely: ACDC (Alternating Columns and Diagonal Centring) developed by Yeredor in 2002 [4], FAJD (Fast Approximative Joint diagonalization Algorithm) developed by Li and Zhang in 2007 [5], UWEDGE (UnWeighted Exhaustive Diagonalization using Gauss itErations) developed by Tichavsky and Yeredor in 2008 [6], JUST (Joint Unitary Shear Transformation) developed by Iferroudjene, Belouchrani and Abed-Meraim in 2009 [7], CVFFdiag (Complex Valued Fast Frobenius diagonalization) developed by Xu, Feng and Zheng in 2011 [8], LUCJD (LU decomposition for Complex Joint Diagonalization) developed by Wang, Gong and Lin in 2012 [9] and CJDi (Complex Joint Diagonalization) developed by Mesloub, AbedMeraim and Belouchrani in 2012 [10]. Note that we restrict our comparative study only to iterative methods and we exclude the non iterative methods [11] [12] which are of higher complexity order especially for large dimensional systems.…”