Abstract
Many source separation algorithms rely on the approximate simultaneous diagonalization of matrices. While there exist very efficient algorithms for symmetric matrices, the skew-symmetric case turned out to be more difficult. Here we show how the often used whitening/rotation approach for symmetric matrices can be translated to this case. While the former leads to orthogonal transformations in Euclidean space, the latter leads to symplectic transformations in symplectic space. It is demonstrated that the resulting algorithm is more stable than a naïve diagonalization that does not respect the symplectic structure of the problem.
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References
Belouchrani, A., Abed-Meraim, K., Cardoso, J.F., Moulines, E.: A blind source separation technique using second order statistics. IEEE Trans. on Signal Processing 45(2), 434–444 (1997)
Cardoso, J.-F., Souloumiac, A.: Blind beamforming for non gaussian signals. IEE Proceedings-F 140, 362–370 (1993)
Meinecke, F.C.: Synchronized? Identifying interactions from superimposed signals. PhD Thesis, TU Berlin (2011)
Meinecke, F.C., Ziehe, A., Nolte, G., Müller, K.-R.: Interacting source analysis - identifying interactions in mixed and noisy complex systems. In: Proc. Int. ICA Research Network 2006, Liverpool, pp. 72–79 (2006)
Nolte, G., Bai, O., Wheaton, L., Mari, Z., Vorbach, S., Hallett, M.: Identifying true brain interaction from eeg data using the imaginary part of coherency. Clinical Neurophysiology 115(10), 2292–2307 (2004)
Nolte, G., Meinecke, F.C., Ziehe, A., Müller, K.-R.: Identifying interactions in mixed and noisy complex systems. Physical Review E 73 (2006)
Plumbley, M.D.: Geometrical methods for non-negative ica: Manifolds, lie groups and toral subalgebras. Neurocomputing 67, 161–197 (2005); Geometrical Methods in Neural Networks and Learning
Ziehe, A., Müller, K.-R.: TDSEP – an efficient algorithm for blind separation using time structure. In: Niklasson, L., Bodén, M., Ziemke, T. (eds.) Proceedings of the 8th International Conference on Artificial Neural Networks, ICANN 1998. Perspectives in Neural Computing, pp. 675–680. Springer, Berlin (1998)
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Meinecke, F.C. (2012). Simultaneous Diagonalization of Skew-Symmetric Matrices in the Symplectic Group. In: Theis, F., Cichocki, A., Yeredor, A., Zibulevsky, M. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2012. Lecture Notes in Computer Science, vol 7191. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28551-6_19
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DOI: https://doi.org/10.1007/978-3-642-28551-6_19
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