Uniqueness Analysis of Non-Unitary Matrix Joint Diagonalization

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 is a joint diagonalizer 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 closed form in terms of an eigen and a singular value decomposition of two matrices.

[1]  Abdelhak M. Zoubir,et al.  Blind separation of nonstationary sources , 2004, IEEE Signal Processing Letters.

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

[3]  Asoke K. Nandi,et al.  Fourth-order cumulant based blind source separation , 1996, IEEE Signal Processing Letters.

[4]  Seungjin Choi,et al.  Independent Component Analysis , 2009, Handbook of Natural Computing.

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

[6]  Hao Shen,et al.  Fast Kernel-Based Independent Component Analysis , 2009, IEEE Transactions on Signal Processing.

[7]  Tülay Adali,et al.  Joint blind source separation by generalized joint diagonalization of cumulant matrices , 2011, Signal Process..

[8]  Tülay Adali,et al.  Blind Separation of Noncircular Correlated Sources Using Gaussian Entropy Rate , 2011, IEEE Transactions on Signal Processing.

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

[10]  Aapo Hyvärinen,et al.  Fast and robust fixed-point algorithms for independent component analysis , 1999, IEEE Trans. Neural Networks.

[11]  Jean-Francois Cardoso,et al.  Source separation using higher order moments , 1989, International Conference on Acoustics, Speech, and Signal Processing,.

[12]  Dinh-Tuan Pham,et al.  Blind separation of instantaneous mixtures of nonstationary sources , 2001, IEEE Trans. Signal Process..

[13]  Arie Yeredor,et al.  Performance Analysis of the Strong Uncorrelating Transformation in Blind Separation of Complex-Valued Sources , 2012, IEEE Transactions on Signal Processing.

[14]  Patrick L. Combettes,et al.  Volterra filtering and higher order whiteness , 1995, IEEE Trans. Signal Process..

[15]  Jean-Franois Cardoso.  High-Order Contrasts for Independent Component Analysis , 1999, Neural Computation.

[16]  Hao Shen,et al.  Complex Blind Source Separation via Simultaneous Strong Uncorrelating Transform , 2010, LVA/ICA.

[17]  Xianda Zhang,et al.  An improved signal-selective direction finding algorithm using second-order cyclic statistics , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[18]  Andreas Ziehe,et al.  An approach to blind source separation based on temporal structure of speech signals , 2001, Neurocomputing.

[19]  K. Abed-Meraim,et al.  Blind source separation using second-order cyclostationary statistics , 2001, 1999 Information, Decision and Control. Data and Information Fusion Symposium, Signal Processing and Communications Symposium and Decision and Control Symposium. Proceedings (Cat. No.99EX251).

[20]  Joos Vandewalle,et al.  Independent component analysis and (simultaneous) third-order tensor diagonalization , 2001, IEEE Trans. Signal Process..

[21]  Jean-Louis Lacoume,et al.  Statistics for complex variables and signals - Part I: Variables , 1996, Signal Process..

[22]  Martin Kleinsteuber,et al.  A sort-Jacobi algorithm for semisimple lie algebras , 2009 .

[23]  R. Liu,et al.  AMUSE: a new blind identification algorithm , 1990, IEEE International Symposium on Circuits and Systems.

[24]  Jerry M. Mendel,et al.  Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications , 1991, Proc. IEEE.

[25]  Jean-Louis Lacoume,et al.  Statistics for complex variables and signals - Part II: signals , 1996, Signal Process..

[26]  Fabian J. Theis,et al.  A New Concept for Separability Problems in Blind Source Separation , 2004, Neural Computation.

[27]  Bart De Moor,et al.  On the blind separation of non-circular sources , 2002, 2002 11th European Signal Processing Conference.

[28]  Bijan Afsari,et al.  Sensitivity Analysis for the Problem of Matrix Joint Diagonalization , 2008, SIAM J. Matrix Anal. Appl..

[29]  Pierre Comon,et al.  Independent component analysis, A new concept? , 1994, Signal Process..

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

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

[32]  Visa Koivunen,et al.  Complex random vectors and ICA models: identifiability, uniqueness, and separability , 2005, IEEE Transactions on Information Theory.

[33]  V. Koivunen,et al.  Ieee Workshop on Machine Learning for Signal Processing Complex-valued Ica Using Second , 2022 .

[34]  J. Cardoso.  On the Performance of Orthogonal Source Separation Algorithms , 1994 .

[35]  Fabian J. Theis,et al.  Uniqueness of complex and multidimensional independent component analysis , 2004, Signal Process..

[36]  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..

[37]  Hao Shen,et al.  Algebraic Solutions to Complex Blind Source Separation , 2012, LVA/ICA.

[38]  Pierre Comon,et al.  Handbook of Blind Source Separation: Independent Component Analysis and Applications , 2010 .

[39]  Karim Abed-Meraim,et al.  Blind source-separation using second-order cyclostationary statistics , 2001, IEEE Trans. Signal Process..

[40]  Hsiao-Chun Wu,et al.  Robust Blind Beamforming Algorithm Using Joint Multiple Matrix Diagonalization , 2005, IEEE Sensors Journal.

[41]  M. B. Priestley,et al.  Non-linear and non-stationary time series analysis , 1990 .

[42]  Vwani P. Roychowdhury,et al.  Independent component analysis based on nonparametric density estimation , 2004, IEEE Transactions on Neural Networks.

[43]  Karim Abed-Meraim,et al.  A general framework for second-order blind separation of stationary colored sources , 2008, Signal Process..

[44]  Riccardo Benedetti,et al.  On simultaneous diagonalization of one Hermitian and one symmetric form , 1984 .

[45]  Hao Shen,et al.  Block Jacobi-type methods for non-orthogonal joint diagonalisation , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[46]  Visa Koivunen,et al.  Complex ICA using generalized uncorrelating transform , 2009, Signal Process..

[47]  Dinh-Tuan Pham,et al.  Mutual information approach to blind separation of stationary sources , 2002, IEEE Trans. Inf. Theory.

[48]  Aapo Hyvärinen,et al.  Blind source separation by nonstationarity of variance: a cumulant-based approach , 2001, IEEE Trans. Neural Networks.

[49]  Michel Verleysen,et al.  Mixing and Non-Mixing Local Minima of the Entropy Contrast for Blind Source Separation , 2006, IEEE Transactions on Information Theory.

[50]  Lang Tong,et al.  Indeterminacy and identifiability of blind identification , 1991 .

[51]  Hao Shen,et al.  Local Convergence Analysis of FastICA and Related Algorithms , 2008, IEEE Transactions on Neural Networks.

[52]  Lucas C. Parra,et al.  Blind Source Separation via Generalized Eigenvalue Decomposition , 2003, J. Mach. Learn. Res..

[53]  Antoine Souloumiac,et al.  Joint diagonalization: Is non-orthogonal always preferable to orthogonal? , 2009, 2009 3rd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[54]  Xianda Zhang,et al.  Direction-of-Arrival Estimation Based on the Joint Diagonalization Structure of Multiple Fourth-Order Cumulant Matrices , 2009, IEEE Signal Processing Letters.

[55]  K. Hüper,et al.  On FastICA Algorithms and Some Generalisations , 2011 .

[56]  Tülay Adali,et al.  A relative gradient algorithm for joint decompositions of complex matrices , 2010, 2010 18th European Signal Processing Conference.