New Jacobi-like algorithms for non-orthogonal joint diagonalization of Hermitian matrices

In this paper, two new algorithms are proposed for non-orthogonal joint matrix diagonalization under Hermitian congruence. The idea of these two algorithms is based on the so-called Jacobi algorithm for solving the eigenvalues problem of Hermitian matrix. The algorithms are then called 'general Jabobi-like diagonalization' algorithms (GERALD). They are based on the search of two complex parameters by the minimization of a quadratic criterion corresponding to a measure of diagonality. Lastly, numerical simulations are conducted to illustrate the effective performances of the GERALD algorithms. HighlightsTwo Jacobi-like algorithms are established under a reasonable assumption.The GERALD2b algorithm converges the fastest among the four algorithms.Convergence statistics are shown to illustrate the performances of algorithms.

[1]  Dimitri Nion,et al.  A Tensor Framework for Nonunitary Joint Block Diagonalization , 2011, IEEE Transactions on Signal Processing.

[2]  P. Tichavsky,et al.  Fast Approximate Joint Diagonalization Incorporating Weight Matrices , 2009, IEEE Transactions on Signal Processing.

[3]  Adel Belouchrani,et al.  A New Algorithm for Complex Non-Orthogonal Joint Diagonalization Based on Shear and Givens Rotations , 2013, IEEE Transactions on Signal Processing.

[4]  Eric Moreau,et al.  Blind Identification and Separation of Complex-Valued Signals: Moreau/Blind Identification and Separation of Complex-Valued Signals , 2013 .

[5]  Hicham Ghennioui,et al.  A Nonunitary Joint Block Diagonalization Algorithm for Blind Separation of Convolutive Mixtures of Sources , 2007, IEEE Signal Processing Letters.

[6]  Eric Moreau,et al.  A Decoupled Jacobi-Like Algorithm for Non-Unitary Joint Diagonalization of Complex-Valued Matrices , 2014, IEEE Signal Processing Letters.

[7]  Xi-Lin Li,et al.  Nonorthogonal Joint Diagonalization Free of Degenerate Solution , 2007, IEEE Transactions on Signal Processing.

[8]  Eric Moreau,et al.  Fast Jacobi like algorithms for joint diagonalization of complex symmetric matrices , 2013, 21st European Signal Processing Conference (EUSIPCO 2013).

[9]  Eric Moulines,et al.  A blind source separation technique using second-order statistics , 1997, IEEE Trans. Signal Process..

[10]  De-Shuang Huang,et al.  Graphical Representation for DNA Sequences via Joint Diagonalization of Matrix Pencil , 2013, IEEE Journal of Biomedical and Health Informatics.

[11]  Lieven De Lathauwer,et al.  A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization , 2006, SIAM J. Matrix Anal. Appl..

[12]  Dinh Tuan Pham,et al.  Joint Approximate Diagonalization of Positive Definite Hermitian Matrices , 2000, SIAM J. Matrix Anal. Appl..

[13]  B. De Moor,et al.  ICA techniques for more sources than sensors , 1999, Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics. SPW-HOS '99.

[14]  Eric Moreau,et al.  Joint Matrices Decompositions and Blind Source Separation , 2014 .

[15]  Christian Jutten,et al.  On the blind source separation of human electroencephalogram by approximate joint diagonalization of second order statistics , 2008, Clinical Neurophysiology.

[16]  El Mostafa Fadaili,et al.  Nonorthogonal Joint Diagonalization/Zero Diagonalization for Source Separation Based on Time-Frequency Distributions , 2007, IEEE Transactions on Signal Processing.

[17]  Shihua Zhu,et al.  Approximate joint diagonalization by nonorthogonal nonparametric Jacobi transformations , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[18]  Arie Yeredor,et al.  Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation , 2002, IEEE Trans. Signal Process..

[19]  Eric Moreau,et al.  A Coordinate Descent Algorithm for Complex Joint Diagonalization Under Hermitian and Transpose Congruences , 2014, IEEE Transactions on Signal Processing.

[20]  Eric Moreau,et al.  High order contrasts for self-adaptive source separation criteria for complex source separation , 1996 .

[21]  Ke Wang,et al.  Complex Non-Orthogonal Joint Diagonalization Based on LU and LQ Decompositions , 2012, LVA/ICA.

[22]  Arie Yeredor,et al.  Joint Matrices Decompositions and Blind Source Separation: A survey of methods, identification, and applications , 2014, IEEE Signal Processing Magazine.

[23]  Antoine Souloumiac,et al.  Nonorthogonal Joint Diagonalization by Combining Givens and Hyperbolic Rotations , 2009, IEEE Transactions on Signal Processing.

[24]  Émilie Chouzenoux,et al.  A block coordinate variable metric forward–backward algorithm , 2016, Journal of Global Optimization.

[25]  Andreas Ziehe,et al.  A Fast Algorithm for Joint Diagonalization with Non-orthogonal Transformations and its Application to Blind Source Separation , 2004, J. Mach. Learn. Res..

[26]  Bijan Afsari,et al.  Some Gradient Based Joint Diagonalization Methods for ICA , 2004, ICA.

[27]  Antoine Souloumiac,et al.  Jacobi Angles for Simultaneous Diagonalization , 1996, SIAM J. Matrix Anal. Appl..

[28]  Bijan Afsari,et al.  Simple LU and QR Based Non-orthogonal Matrix Joint Diagonalization , 2006, ICA.

[29]  J. Cardoso,et al.  Blind beamforming for non-gaussian signals , 1993 .

[30]  Eric Moreau,et al.  Variations around gradient like algorithms for joint diagonalization of hermitian matrices , 2012, 2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO).

[31]  Eric Moreau,et al.  A one stage self-adaptive algorithm for source separation , 1994, Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing.

[32]  A. Bunse-Gerstner,et al.  Numerical Methods for Simultaneous Diagonalization , 1993, SIAM J. Matrix Anal. Appl..

[33]  Eric Moreau,et al.  A generalization of joint-diagonalization criteria for source separation , 2001, IEEE Trans. Signal Process..

[34]  Zheng Bao,et al.  An efficient multistage decomposition approach for independent components , 2003, Signal Process..