Complex random vectors and ICA models: identifiability, uniqueness, and separability

In this paper, the conditions for identifiability, separability and uniqueness of linear complex valued independent component analysis (ICA) models are established. These results extend the well-known conditions for solving real-valued ICA problems to complex-valued models. Relevant properties of complex random vectors are described in order to extend the Darmois-Skitovich theorem for complex-valued models. This theorem is used to construct a proof of a theorem for each of the above ICA model concepts. Both circular and noncircular complex random vectors are covered. Examples clarifying the above concepts are presented

[1]  R. Wooding.  The multivariate distribution of complex normal variables , 1956 .

[2]  Z. Maguar,et al.  Functional equations in several variables , 1993 .

[3]  James L. Massey,et al.  Proper complex random processes with applications to information theory , 1993, IEEE Trans. Inf. Theory.

[4]  I︠u︡. V. Linnik,et al.  Decomposition of Random Variables and Vectors , 1977 .

[5]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[6]  Arak M. Mathai,et al.  Characterizations of the normal probability law , 1977 .

[7]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[8]  Vince D. Calhoun,et al.  Complex Infomax: Convergence and Approximation of Infomax with Complex Nonlinearities , 2002, Proceedings of the 12th IEEE Workshop on Neural Networks for Signal Processing.

[9]  Jerry Alan Veeh,et al.  Complex Stable Sums of Complex Stable Random Variables , 2001 .

[10]  Pascal Bondon,et al.  Second-order statistics of complex signals , 1997, IEEE Trans. Signal Process..

[11]  J. Aczél,et al.  Lectures on Functional Equations and Their Applications , 1968 .

[12]  Aapo Hyvärinen,et al.  A Fast Fixed-Point Algorithm for Independent Component Analysis of Complex Valued Signals , 2000, Int. J. Neural Syst..

[13]  Georg Taubock.  Rotationally Variant Complex Channels , 2002 .

[14]  B. L. S. Prakasa Rao,et al.  On a characterization of multivariate normal distribution , 1974 .

[15]  Simone Fiori,et al.  Extended Hebbian learning for blind separation of complex-valued sources , 2003, IEEE Trans. Circuits Syst. II Express Briefs.

[16]  I. Ostrovskiǐ,et al.  The Arithmetic of Probability Distributions , 1977 .

[17]  Louis L. Scharf,et al.  Second-order analysis of improper complex random vectors and processes , 2003, IEEE Trans. Signal Process..

[18]  Николай Николаевич Вахания,et al.  Случайные векторы со значениями в комплексных гильбертовых пространствах@@@Random vectors with values in complex Hilbert spaces , 1996 .

[19]  N. R. Goodman.  Statistical analysis based on a certain multivariate complex Gaussian distribution , 1963 .

[20]  Terrence J. Sejnowski,et al.  Complex Independent Component Analysis of Frequency-Domain Electroencephalographic Data , 2003, Neural Networks.

[21]  Gennadii Feldman.  Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups , 1993 .

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

[23]  Roger Cuppens,et al.  Decomposition of multivariate probabilities , 1975 .

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

[25]  J. Cardoso,et al.  An efficient technique for the blind separation of complex sources , 1993, [1993 Proceedings] IEEE Signal Processing Workshop on Higher-Order Statistics.

[26]  Pascal Chevalier,et al.  Widely linear estimation with complex data , 1995, IEEE Trans. Signal Process..

[27]  P. Krishnaiah,et al.  Complex elliptically symmetric distributions , 1986 .

[28]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[29]  Andrzej Cichocki,et al.  Adaptive Blind Signal and Image Processing - Learning Algorithms and Applications , 2002 .

[30]  Bernard C. Picinbono,et al.  Second-order complex random vectors and normal distributions , 1996, IEEE Trans. Signal Process..

[31]  N. N. Vakhania,et al.  Random Vectors with Values in Complex Hilbert Spaces , 1997 .

[32]  Pierre Comon,et al.  Analytical blind channel identification , 2002, IEEE Trans. Signal Process..

[33]  Visa Koivunen,et al.  Identifiability, separability, and uniqueness of linear ICA models , 2004, IEEE Signal Processing Letters.

[34]  Herbert Heyer.  Book Review: Arithmetic of probability distributions and characterization problems on Abelian groups , 1995 .

[35]  N. Vakhania,et al.  Probability Distributions on Banach Spaces , 1987 .

[36]  Nicholas N. Vakhania,et al.  Polya's Characterization Theorem for Complex Random Variables , 1997, J. Complex..