Elsevier

Signal Processing

Volume 84, Issue 2, February 2004, Pages 217-229
Signal Processing

Three easy ways for separating nonlinear mixtures?

https://doi.org/10.1016/j.sigpro.2003.10.011Get rights and content

Abstract

In this paper, we consider the nonlinear Blind Source Separation BSS and independent component analysis (ICA) problems, and especially uniqueness issues, presenting some new results. A fundamental difficulty in the nonlinear BSS problem and even more so in the nonlinear ICA problem is that they are nonunique without a suitable regularization. In this paper, we mainly discuss three different ways for regularizing the solutions, that have been recently explored.

Section snippets

Model and problem

Consider N samples of the m-dimension observed random vector x, modeled byx=F(s)+n,where F is an unknown mixing mapping assumed invertible, s is an unknown n-dimensional source vector containing the source signals s1,s2,…,sn, which are assumed to be statistically independent, and n is an additive noise, independent of the sources.

Such a model is usual in multidimensional signal processing, where each sensor receives an unknown superimposition of unknown source signals at time instants t=1,…,N.

Existence and uniqueness of nonlinear ICA and BSS

Several authors [18], [24], [26], [39], [40] have recently addressed the important issues on the existence and uniqueness of solutions for the nonlinear ICA and BSS problems. Their main results, which are direct consequences of Darmois's results on factorial analysis [16], are reported in this section.

Smooth mappings

Recently, multi-layer perceptron (MLP) networks (see [19]) have been used in [2], [48] for estimating the generic nonlinear mappings H. Especially, Almeida conjectured that smooth mappings providing by MLP networks leads to a regularization sufficient for ensuring that nonlinear ICA leads to nonlinear BSS, too. However, the following example [5] shows that smoothness alone is not sufficient for separation.

Without a loss of generality, consider two independent random variables s=(s1,s2)T which

Structural constraints

A natural way of regularizing the solution consists in looking for separating mappings belonging to a specific subspace Q. To characterize the indeterminacies for this specific model Q, one must solve the tricky independence preservation equation which can be written∀E∈Mn,EdFs1dFs2dFsn=H(E)dFy1dFy2dFyn,where Mn is the set of all the measurable compacts in Rn (in other words, Mn is a σ-algebra on Rn), and Fsi denotes the distribution function of the random variable si .

Let P denote the set2

Prior information on the sources

In this section we show that prior information on the sources can simplify or relax the indeterminacies. The first example takes into account that sources are bounded. The second example exploits the temporal correlation of the sources.

Concluding remarks

In this paper, we have considered ICA and BSS problems for nonlinear mixture models. It appears clearly BSS and ICA are difficult and ill-posed problems, and regularization is necessary for actually achieving ICA solutions which coincide to BSS.

In this purpose, two main ways can be used. First, solving the nonlinear BSS problem appropriately using only the independence assumption is possible only if mixtures as well as separation structure are structurally constrained: for example

Acknowledgements

This work has been supported by the European Commission project BLISS (IST-1999-14190). The authors thank Dr. A. Taleb for contributing many of the results presented in this paper, and the three anonymous reviewers for their very relevant and detailed comments which contributed to improve this paper.

References (49)

  • M. Babaie-Zadeh, On blind source separation in convolutive and nonlinear mixtures, Ph.D. Thesis, INPG, Grenoble,...
  • M. Babaie-Zadeh, C. Jutten, K. Nayebi, Separating convolutive post non-linear mixtures, in: Proceedings of the Third...
  • M. Babaie-Zadeh, C. Jutten, K. Nayebi, A geometric approach for separating post nonlinear mixtures, in: Proceedings of...
  • A. Bell et al.

    An information-maximization approach to blind separation and blind deconvolution

    Neural Computation

    (1995)
  • A. Belouchrani et al.

    A blind source separation technique based on second order statistics

    IEEE Trans. Signal Processing

    (1997)
  • J.-F. Cardoso

    Blind signal separation: statistical principles

    Proc. IEEE

    (1998)
  • J.-F. Cardoso et al.

    Equivariant adaptive source separation

    IEEE Trans. Signal Processing

    (1996)
  • J.-F. Cardoso et al.

    Blind beamforming for non gaussian signals

    IEE Proc.-F

    (1993)
  • A. Cichocki et al.

    Adaptive Blind Signal and Image Processing—Learning Algorithms and Applications

    (2002)
  • A. Cichocki et al.

    Robust learning algorithm for blind separation of signals

    Electron. Lett.

    (1994)
  • G. Darmois, Analyse des liaisons de probabilité, in: Proceedings of International Statistics Conferences 1947, Vol. III...
  • J. Eriksson, V. Koivunen, Blind identifiability of class of nonlinear instantaneous ICA models, in: Proceedings of the...
  • S. Haykin

    Neural Networks—A Comprehensive Foundation

    (1998)
  • S. Hosseini et al.

    On the separability of nonlinear mixtures of temporally correlated sources

    IEEE Signal Processing Lett.

    (2003)
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