Mixing and non-mixing local minima of the entropy contrast for blind source separation

In this paper, both non-mixing and mixing local minima of the entropy are analyzed from the viewpoint of blind source separation (BSS); they correspond respectively to acceptable and spurious solutions of the BSS problem. The contribution of this work is twofold. First, a Taylor development is used to show that the \textit{exact} output entropy cost function has a non-mixing minimum when this output is proportional to \textit{any} of the non-Gaussian sources, and not only when the output is proportional to the lowest entropic source. Second, in order to prove that mixing entropy minima exist when the source densities are strongly multimodal, an entropy approximator is proposed. The latter has the major advantage that an error bound can be provided. Even if this approximator (and the associated bound) is used here in the BSS context, it can be applied for estimating the entropy of any random variable with multimodal density.Comment: 11 pages, 6 figures, To appear in IEEE Transactions on Information Theor

Information Theoretic Proofs of Entropy Power Inequalities

While most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information, up to now Shannon's entropy power inequality (EPI) is an exception: Existing information theoretic proofs of the EPI hinge on representations of differential entropy using either Fisher information or minimum mean-square error (MMSE), which are derived from de Bruijn's identity. In this paper, we first present an unified view of these proofs, showing that they share two essential ingredients: 1) a data processing argument applied to a covariance-preserving linear transformation; 2) an integration over a path of a continuous Gaussian perturbation. Using these ingredients, we develop a new and brief proof of the EPI through a mutual information inequality, which replaces Stam and Blachman's Fisher information inequality (FII) and an inequality for MMSE by Guo, Shamai and Verd\'u used in earlier proofs. The result has the advantage of being very simple in that it relies only on the basic properties of mutual information. These ideas are then generalized to various extended versions of the EPI: Zamir and Feder's generalized EPI for linear transformations of the random variables, Takano and Johnson's EPI for dependent variables, Liu and Viswanath's covariance-constrained EPI, and Costa's concavity inequality for the entropy power.Comment: submitted for publication in the IEEE Transactions on Information Theory, revised versio

Mixing and Non-Mixing Local Minima of the Entropy Contrast for Blind Source Separation

Abstract—In this paper, both non-mixing and mixing local minima of the entropy are analyzed from the viewpoint of blind source separation (BSS); they correspond respectively to acceptable and spurious solutions of the BSS problem. The contribution of this work is twofold. First, a Taylor development is used to show that the exact output entropy cost function has a non-mixing minimum when this output is proportional to any of the non-Gaussian sources, and not only when the output is proportional to the lowest entropic source. Second, in order to prove that mixing entropy minima exist when the source densities are strongly multimodal, an entropy approximator is proposed. The latter has the major advantage that an error bound can be provided. Even if this approximator (and the associated bound) is used here in the BSS context, it can be applied for estimating the entropy of any random variable with multimodal density. Index Terms—Blind source separation (BSS), entropy estimation, independent component analysis, mixture distribution, multimodal densities. I