Abstract
We introduce the terms strong sub- and super-Gaussianity to refer to the previously introduced class of densities log-concave is x 2 and log-convex in x 2 respectively. We derive relationships among the various definitions of sub- and super-Gaussianity, and show that strong sub- and super-Gaussianity are related to the score function being star-shaped upward or downward with respect to the origin. We illustrate the definitions and results by extending a theorem of Benveniste, Goursat, and Ruget on uniqueness of separating local optima in ICA.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Amari, S.-I., Chen, T.-P., Cichocki, A.: Stability analysis of learning algorithms for blind source separation. Neural Networks 10(8), 1345–1351 (1997)
Andrews, D.F., Mallows, C.L.: Scale mixtures of normal distributions. J. Roy. Statist. Soc. Ser. B 36, 99–102 (1974)
Benveniste, A., Goursat, M., Ruget, G.: Robust identification of a nonminimum phase system. IEEE Transactions on Automatic Control 25(3), 385–399 (1980)
Benveniste, A., Métivier, M., Priouret, P.: Adaptive algorithms and stochastic approximations. Springer, Heidelberg (1990)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Bruckner, A.M., Ostrow, E.: Some function classes related to the class of convex functions. Pacific J. Math. 12(4), 1203–1215 (1962)
Cardoso, J.-F., Laheld, B.H.: Equivariant adaptive source separation. IEEE. Trans. Sig. Proc. 44(12), 3017–3030 (1996)
Comon, P., Jutten, C. (eds.): Handbook of Blind Source Separation: Independent Component Analysis and Applications. Elsevier, Amsterdam (2010)
Finucan, H.M.: A note on kurtosis. Journal of the Royal Statistical Society, Series B (Methodological) 26(1), 111–112 (1964)
Haykin, S.: Neural Networks: a comprehensive foundation. Prentice-Hall, Englewood Cliffs (1999)
Karlin, S., Novikoff, A.: Generalized convex inequalities. Pacific J. Math. 13, 1251–1279 (1963)
Keilson, J., Steutel, F.W.: Mixtures of distributions, moment inequalities, and measures of exponentiality and Normality. The Annals of Probability 2, 112–130 (1974)
Mansour, A., Jutten, C.: What should we say about the kurtosis? IEEE Signal Processing Letters 6(12), 321–322 (1999)
Palmer, J.A., Kreutz-Delgado, K., Wipf, D.P., Rao, B.D.: Variational EM algorithms for non-gaussian latent variable models. In: Advances in Neural Information Processing Systems. MIT Press, Cambridge (2006)
Rao, B.D., Engan, K., Cotter, S.F., Palmer, J., Kreutz-Delgado, K.: Subset selection in noise based on diversity measure minimization. IEEE Trans. Signal Processing 51(3) (2003)
Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)
Williamson, R.: Multiply monotone functions and their Laplace transforms. Duke Math. J. 23, 189–207 (1956)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Palmer, J.A., Kreutz-Delgado, K., Makeig, S. (2010). Strong Sub- and Super-Gaussianity. In: Vigneron, V., Zarzoso, V., Moreau, E., Gribonval, R., Vincent, E. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2010. Lecture Notes in Computer Science, vol 6365. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15995-4_38
Download citation
DOI: https://doi.org/10.1007/978-3-642-15995-4_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15994-7
Online ISBN: 978-3-642-15995-4
eBook Packages: Computer ScienceComputer Science (R0)