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On the Optimal Non-linearities for Gaussian Mixtures in FastICA

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Latent Variable Analysis and Signal Separation (LVA/ICA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10169))

Abstract

In independent component analysis we assume that the observed vector is a linear transformation of a latent vector of independent components, our objective being the estimation of the latter. Deflation-based FastICA estimates the components one-by-one by repeatedly maximizing the expected value of some function measuring non-Gaussianity, the derivative of which is called the non-linearity. Under some weak assumptions, the asymptotically optimal non-linearity for extracting sources with a specific density is given by the location score function of the density. In this paper we look into the consequences of this result from the viewpoint of estimating Gaussian location and scale mixtures. As one of our results we justify the common use of hyperbolic tangent, tanh, as a non-linearity in blind clustering by showing that it is optimal for estimating certain Gaussian mixtures. Finally, simulations are used to show that the asymptotic optimality results hold in various settings also for finite samples.

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Acknowledgements

We would like to thank the anonymous referees for their stimulating comments which enhanced the paper and provided us with existing results previously unknown to us. This work was supported by the Academy of Finland Grant 268703.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Turku, Turku, Finland

    Joni Virta & Klaus Nordhausen

  2. School of Health Sciences, University of Tampere, Tampere, Finland

    Klaus Nordhausen

Authors
  1. Joni Virta
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  2. Klaus Nordhausen
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Corresponding author

Correspondence to Joni Virta .

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Editors and Affiliations

  1. Institute of Information Theory and Automation, Prague, Czech Republic

    Petr Tichavský

  2. Sharif University of Technology, Tehran, Iran

    Massoud Babaie-Zadeh

  3. Grenoble-Alpes University, Grenoble, France

    Olivier J.J. Michel

  4. Toulon University, Toulon, France

    Nadège Thirion-Moreau

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Virta, J., Nordhausen, K. (2017). On the Optimal Non-linearities for Gaussian Mixtures in FastICA. In: Tichavský, P., Babaie-Zadeh, M., Michel, O., Thirion-Moreau, N. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2017. Lecture Notes in Computer Science(), vol 10169. Springer, Cham. https://doi.org/10.1007/978-3-319-53547-0_40

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  • DOI: https://doi.org/10.1007/978-3-319-53547-0_40

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53546-3

  • Online ISBN: 978-3-319-53547-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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