Second Order Subspace Analysis and Simple Decompositions

  • Harold W. Gutch
  • Takanori Maehara
  • Fabian J. Theis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6365)

Abstract

The recovery of the mixture of an N-dimensional signal generated by N independent processes is a well studied problem (see e.g. [1,10]) and robust algorithms that solve this problem by Joint Diagonalization exist. While there is a lot of empirical evidence suggesting that these algorithms are also capable of solving the case where the source signals have block structure (apart from a final permutation recovery step), this claim could not be shown yet - even more, it previously was not known if this model separable at all. We present a precise definition of the subspace model, introducing the notion of simple components, show that the decomposition into simple components is unique and present an algorithm handling the decomposition task.

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References

  1. 1.
    Belouchrani, A., Abed-Meraim, K., Cardoso, J.-F., Moulines, E.: A blind source separation technique using second-order statistics. IEEE Transactions on Signal Processing 45(2), 434–444 (1997)CrossRefGoogle Scholar
  2. 2.
    Bunse-Gerstner, A., Byers, R., Mehrmann, V.: Numerical methods for simultaneous diagonalization. SIAM J. Matrix Anal. Appl. 14(4), 927–949 (1993)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cardoso, J.-F., Souloumiac, A.: Jacobi angles for simultaneous diagonalization. SIAM J. Matrix Anal. Appl. 17(1), 161–164 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gutch, H.W., Theis, F.J.: Independent subspace analysis is unique, given irreducibility. In: Davies, M.E., James, C.J., Abdallah, S.A., Plumbley, M.D. (eds.) ICA 2007. LNCS, vol. 4666, pp. 49–56. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Liu, W., Mandic, D.P., Cichocki, A.: Blind source extraction based on a linear predictor. IET Signal Process. 1(1), 29–34 (2007)CrossRefGoogle Scholar
  6. 6.
    Maehara, T., Murota, K.: Error-controlling algorithm for simultaneous block-diagonalization and its application to independent component analysis. JSIAM Letters (submitted)Google Scholar
  7. 7.
    Maehara, T., Murota, K.: Algorithm for error-controlled simultaneous block-diagonalization of matrices. Technical Report METR-2009-53 (December 2009)Google Scholar
  8. 8.
    Molgedey, L., Schuster, H.G.: Separation of a mixture of independent signals using time delayed correlations. Phys. Rev. Lett. 72(23), 3634–3637 (1994)CrossRefGoogle Scholar
  9. 9.
    Theis, F.J.: Towards a general independent subspace analysis. In: Proc. NIPS, pp. 1361–1368 (January 2006)Google Scholar
  10. 10.
    Tong, L., Soon, V.C., Huang, Y.-F., Liu, R.: AMUSE: a new blind identification algorithm. In: IEEE International Symposium on Circuits and Systems, vol. 3, pp. 1784–1787 (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Harold W. Gutch
    • 1
    • 2
  • Takanori Maehara
    • 3
  • Fabian J. Theis
    • 2
    • 4
  1. 1.Department of Nonlinear DynamicsMax Planck Institute for Dynamics and Self-OrganizationGermany
  2. 2.Technical University of MunichGermany
  3. 3.University of TokyoJapan
  4. 4.Helmholtz-Institute NeuherbergGermany

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