On Using Quaternionic Rotations for Indpendent Component Analysis

Independent component analysis (ICA) is a popular technique for demixing multi-sensor data. In many approaches to the ICA, signals are decorrelated by whitening data and then by rotating the result. In this paper, we introduce a four-unit, symmetric algorithm, based on quaternionic factorization of rotation matrix. It makes use an isomorphism between quaternions and $4\times 4$ orthogonal matrices. Unlike conventional techniques based on Jacobi decomposition, our method exploits 4D rotations and uses negentropy approximation as a contrast function. Compared to the widely used, symmetric FastICA algorithm, the proposed method offers a better separation quality in a presence of multiple Gaussian sources.

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

[2]  A. Hyvarinen,et al.  One-unit contrast functions for independent component analysis: a statistical analysis , 1997, Neural Networks for Signal Processing VII. Proceedings of the 1997 IEEE Signal Processing Society Workshop.

[3]  Aapo Hyvärinen,et al.  The Fixed-Point Algorithm and Maximum Likelihood Estimation for Independent Component Analysis , 1999, Neural Processing Letters.

[4]  Aapo Hyvärinen,et al.  New Approximations of Differential Entropy for Independent Component Analysis and Projection Pursuit , 1997, NIPS.

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

[6]  Sun-Yuan Kung,et al.  On gradient adaptation with unit-norm constraints , 2000, IEEE Trans. Signal Process..

[7]  T. Howell,et al.  The Complexity of the Quaternion Product , 1975 .

[8]  E. Oja,et al.  Performance Analysis of the FastICA Algorithm and Cramér – Rao Bounds for Linear Independent Component Analysis , 2010 .

[9]  Andreas Polze,et al.  Fast ICA on Modern GPU Architectures , 2014, 2014 15th International Conference on Parallel and Distributed Computing, Applications and Technologies.

[10]  Erkki Oja,et al.  Independent component analysis: algorithms and applications , 2000, Neural Networks.

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

[12]  Niloufer Mackey,et al.  Hamilton and Jacobi Meet Again: Quaternions and the Eigenvalue Problem , 1995, SIAM J. Matrix Anal. Appl..

[13]  Pierre Comon,et al.  Robust Independent Component Analysis by Iterative Maximization of the Kurtosis Contrast With Algebraic Optimal Step Size , 2010, IEEE Transactions on Neural Networks.

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

[15]  Erkki Oja,et al.  Performance analysis of the FastICA algorithm and Crame/spl acute/r-rao bounds for linear independent component analysis , 2006, IEEE Transactions on Signal Processing.