Fast Approximate Joint Diagonalization of Positive Definite Hermitian Matrices

In this paper, a new efficient iterative algorithm for approximate joint diagonalization of positive-definite Hermitian matrices is presented. The proposed algorithm, named as SVDJD, estimates the diagonalization matrix by iterative optimization of a maximum likelihood based objective function. The columns of the diagonalization matrix is not assumed to be orthogonal, and they are estimated separately by using iterative singular value decompositions of a weighted sum of the matrices to be diagonalized. The performance of the proposed SVDJD algorithm is evaluated and compared to other existing state-of-the-art algorithms for approximate joint diagonalization. The results imply that the SVDJD algorithm is computationally efficient with performance similar to state-of-the-art algorithms for approximate joint diagonalization.

[1]  Arogyaswami Paulraj,et al.  An analytical constant modulus algorithm , 1996, IEEE Trans. Signal Process..

[2]  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..

[3]  S. Amari.  Differential Geometry of Statistical Models , 1985 .

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

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

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

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

[8]  F. Götze.  Differential-geometrical methods in statistics. Lecture notes in statistics - A. Shun-ichi. , 1987 .

[9]  M. Joho,et al.  Joint diagonalization of correlation matrices by using Newton methods with application to blind signal separation , 2002, Sensor Array and Multichannel Signal Processing Workshop Proceedings, 2002.

[10]  Joseph Tabrikian,et al.  Blind Separation of Independent Sources Using Gaussian Mixture Model , 2007, IEEE Transactions on Signal Processing.

[11]  W. Gautschi,et al.  An algorithm for simultaneous orthogonal transformation of several positive definite symmetric matrices to nearly diagonal form , 1986 .

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

[13]  T. Ens,et al.  Blind signal separation : statistical principles , 1998 .

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

[15]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[16]  Alle-Jan van der Veen,et al.  Joint diagonalization via subspace fitting techniques , 2001, ICASSP.