Identifiability of Complex Blind Source Separation via Non-Unitary Joint Diagonalization

Identiability analysis of complex Blind Source Separation (BSS), i.e. to study under what conditions the BSS problem can be solved, is a long- standing and most critical problem in the community. It serves not only as the indicator to solvability of the BSS problem, but also as the con- structive ground for developing ecient algorithms. Various BSS methods are based on jointly diagonalizing a set of matrices, which are generated using second- or higher-order statistics. The present work provides a gen- eral result on the uniqueness conditions of matrix joint diagonalization. It unies all existing results on the identiability conditions of complex BSS, with respect to non-circularity, non-stationarity, non-whiteness, and non-Gaussianity. Additionally, following the main identiability result, a solution for complex BSS is proposed. It is given in closed form in terms of an eigenvalue and a singular value decomposition of two matrices. Index Terms Complex Blind Source Separation (BSS), Second-Order Statistics (SOS), Higher-Order Statistics (HOS), non-unitary joint diagonalization, Iwasawa decomposition.

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

[2]  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.

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

[4]  Jose C. Principe,et al.  Robust Blind Beamforming Algorithm Using Joint Multiple Matrix Diagonalization , 2007 .

[5]  E. Oja,et al.  Independent Component Analysis , 2001 .

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

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

[8]  R. Horn,et al.  An analog of the singular value decomposition for complex orthogonal equivalence , 1987 .

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

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

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

[12]  Visa Koivunen,et al.  Complex ICA using generalized uncorrelating transform , 2009, 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]  Hao Shen,et al.  Complex Blind Source Separation via Simultaneous Strong Uncorrelating Transform , 2010, LVA/ICA.

[15]  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.

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

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

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

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

[20]  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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[38]  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.

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

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

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

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

[43]  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..

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

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

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