CONFAC Decomposition Approach to Blind Identification of Underdetermined Mixtures Based on Generating Function Derivatives

• Published in: IEEE transactions on signal processing Vol. 60; no. 11; pp. 5698 - 5713
• Main Authors: de Almeida, A. L. F., Luciani, X., Stegeman, A., Comon, P.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.11.2012; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Applied sciences; Blind identification; Blinds; complex sources; Complexity; Complexity theory; Computer Science; CONFAC decomposition; Constraints; Decomposition; Derivatives; Detection, estimation, filtering, equalization, prediction; Engineering Sciences; Estimation; Exact sciences and technology; Information, signal and communications theory; Least squares approximation; Mathematical analysis; Mathematical models; Matrix decomposition; second generating function; Signal and communications theory; Signal and Image Processing; Signal, noise; Studies; Telecommunications and information theory; Tensile stress; Vectors; Performance evaluation; Second order; Generating function; Third order; CONFAC decomposition; Blind identification; Computational complexity; Search algorithm; Complex variable method; Sufficient condition; second generating function; Simulation; Levenberg Marquardt algorithm; Least squares method; Signal processing; complex sources; Function derivative; System identification; Cumulant; tensor
• ISSN: 1053-587X; 1941-0476; 1941-0476
• DOI: 10.1109/TSP.2012.2208956

This work proposes a new tensor-based approach to solve the problem of blind identification of underdetermined mixtures of complex-valued sources exploiting the cumulant generating function (CGF) of the observations. We show that a collection of second-order derivatives of the CGF of the observations can be stored in a third-order tensor following a constrained factor (CONFAC) decomposition with known constrained structure. In order to increase the diversity, we combine three derivative types into an extended third-order CONFAC decomposition. A detailed uniqueness study of this decomposition is provided, from which easy-to-check sufficient conditions ensuring the essential uniqueness of the mixing matrix are obtained. From an algorithmic viewpoint, we develop a CONFAC-based enhanced line search (CONFAC-ELS) method to be used with an alternating least squares estimation procedure for accelerated convergence, and also analyze the numerical complexities of two CONFAC-based algorithms (namely, CONFAC-ALS and CONFAC-ELS) in comparison with the Levenberg-Marquardt (LM)-based algorithm recently derived to solve the same problem. Simulation results compare the proposed approach with some higher-order methods. Our results also corroborate the advantages of the CONFAC-based approach over the competing LM-based approach in terms of performance and computational complexity.