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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References
Cardoso, J.F., Souloumiac, A.: Blind beamforming for non-Gaussian signals. IEE Proc. F Radar Sig. Process. 140, 362–370 (1993)
Dermoune, A., Wei, T.: FastICA algorithm: five criteria for the optimal choice of the nonlinearity function. IEEE Trans. Sig. Process. 61(8), 2078–2087 (2013)
Gómez-Sánchez-Manzano, E., Gómez-Villegas, M., Marín, J.: Sequences of elliptical distributions and mixtures of normal distributions. J. Multivar. Anal. 97(2), 295–310 (2006)
Huber, P.J.: Projection pursuit. Ann. Stat. 13(2), 435–475 (1985)
Hyvärinen, A.: One-unit contrast functions for independent component analysis: a statistical analysis. In: Proceedings of the 1997 IEEE Workshop on Neural Networks for Signal Processing, pp. 388–397 (1997)
Hyvärinen, A.: Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. Neural Netw. 10(3), 626–634 (1999)
Hyvärinen, A., Karhunen, J., Oja, E.: Independent Component Analysis. Wiley, New York (2001)
Hyvärinen, A., Oja, E.: A fast fixed-point algorithm for independent component analysis. Neural Comput. 9, 1483–1492 (1997)
Koldovskỳ, Z., Tichavskỳ, P., Oja, E.: Efficient variant of algorithm FastICA for independent component analysis attaining the Cramer-Rao lower bound. IEEE Trans. Neural Netw. 17(5), 1265–1277 (2006)
Miettinen, J., Nordhausen, K., Oja, H., Taskinen, S.: Deflation-based FastICA with adaptive choices of nonlinearities. IEEE Trans. Sig. Process. 62(21), 5716–5724 (2014)
Miettinen, J., Nordhausen, K., Oja, H., Taskinen, S., Virta, J.: The squared symmetric FastICA estimator. Sig. Process. 131, 402–411 (2017)
Miettinen, J., Taskinen, S., Nordhausen, K., Oja, H.: Fourth moments and independent component analysis. Stat. Sci. 30, 372–390 (2015)
Nordhausen, K., Ilmonen, P., Mandal, A., Oja, H., Ollila, E.: Deflation-based FastICA reloaded. In: Proceedings of 19th European Signal Processing Conference, pp. 1854–1858 (2011)
Ollila, E.: The deflation-based FastICA estimator: statistical analysis revisited. IEEE Trans. Sig. Process. 58(3), 1527–1541 (2010)
Palmer, J., Kreutz-Delgado, K., Rao, B.D., Wipf, D.P.: Variational EM algorithms for non-gaussian latent variable models. In: Advances in Neural Information Processing Systems, pp. 1059–1066 (2005)
Tichavský, P., Koldovský, Z., Oja, E.: Speed and accuracy enhancement of linear ICA techniques using rational nonlinear functions. In: Davies, M.E., James, C.J., Abdallah, S.A., Plumbley, M.D. (eds.) ICA 2007. LNCS, vol. 4666, pp. 285–292. Springer, Heidelberg (2007). doi:10.1007/978-3-540-74494-8_36
Virta, J., Nordhausen, K., Oja, H.: Projection pursuit for non-Gaussian independent components. arXiv (2016). https://arxiv.org/abs/1612.05445
Wei, T.: On the spurious solutions of the FastICA algorithm. In: IEEE Workshop on Statistical Signal Processing, pp. 161–164 (2014)
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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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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