New ICA Algorithms based on Special linear Group

Unlike the traditional independent component analysis (ICA) algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices, in this paper we propose two new methods which only search for solutions in the space of the matrices with unitary determinant and without whitening. The new algorithms are based on the special linear group SL(n). In order to achieve our target, we first provide a representation theory for any matrix in SL(n), which only simply uses the product of multiple exponentials of traceless matrices. Based on the matrix representation theory, two novel ICA algorithms are developed along with simple analysis on their equilibrium points. Moreover, we apply our methods to the classical problem of signal separation. The experimental results indicate that the superior convergence of our proposed algorithms, which can be expected as two viable alternatives to the ICA algorithms available in publications.

[1]  Ze Wang.  Fixed-point algorithms for constrained ICA and their applications in fMRI data analysis. , 2011, Magnetic resonance imaging.

[2]  Sabine Van Huffel,et al.  Spatially constrained ICA algorithm with an application in EEG processing , 2011, Signal Process..

[3]  Tülay Adali,et al.  Complex ICA by Negentropy Maximization , 2008, IEEE Transactions on Neural Networks.

[4]  Hualiang Li,et al.  Complex ICA Using Nonlinear Functions , 2008, IEEE Transactions on Signal Processing.

[5]  Marc M. Van Hulle,et al.  Sequential Fixed-Point ICA Based on Mutual Information Minimization , 2008, Neural Computation.

[6]  Shotaro Akaho,et al.  Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold , 2005, Neurocomputing.

[7]  Nicoletta Del Buono,et al.  A Survey on Methods for Computing Matrix Exponentials in Numerical Schemes for ODEs , 2003, International Conference on Computational Science.

[8]  Mark D. Plumbley.  Geometrical methods for non-negative ICA: Manifolds, Lie groups and toral subalgebras , 2005, Neurocomputing.

[9]  Kazunori Yamaguchi,et al.  Linear Multilayer ICA Using Adaptive PCA , 2009, Neural Processing Letters.

[10]  Simone G. O. Fiori,et al.  Learning independent components on the orthogonal group of matrices by retractions , 2007, Neural Processing Letters.

[11]  Jean-François Cardoso,et al.  Equivariant adaptive source separation , 1996, IEEE Trans. Signal Process..

[12]  Simone G. O. Fiori,et al.  Quasi-Geodesic Neural Learning Algorithms Over the Orthogonal Group: A Tutorial , 2005, J. Mach. Learn. Res..

[13]  P. Absil,et al.  Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation , 2004 .

[14]  P. Tichavsky,et al.  Fast Approximate Joint Diagonalization Incorporating Weight Matrices , 2009, IEEE Transactions on Signal Processing.

[15]  Dinesh Kumar,et al.  Estimation of independent and dependent components of non-invasive EMG using fast ICA: validation in recognising complex gestures , 2011, Computer methods in biomechanics and biomedical engineering.

[16]  Jonathan H. Manton,et al.  Optimization algorithms exploiting unitary constraints , 2002, IEEE Trans. Signal Process..

[17]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[18]  Hualiang Li,et al.  Algorithms for Complex ML ICA and Their Stability Analysis Using Wirtinger Calculus , 2010, IEEE Transactions on Signal Processing.

[19]  Francesco Piazza,et al.  Stability analysis of natural gradient learning rules in overdetermined ICA , 2008, Signal Process..