Ica_spurious_vrins05 - 190 IEEE SIGNAL PROCESSING LETTERS VOL 12 NO 3 MARCH 2005 Information Theoretic versus Cumulant-Based Contrasts for

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Unformatted text preview: 190 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 3, MARCH 2005 Information Theoretic versus Cumulant-Based Contrasts for Multimodal Source Separation Frédéric Vrins and Michel Verleysen , Senior Member, IEEE Abstract— Recently, several authors have emphasized the exis- tence of spurious maxima in usual contrast functions for source separation (e.g., the likelihood and the mutual information) when several sources have multimodal distributions. The aim of this letter is to compare the information theoretic contrasts to cumu- lant-based ones from the robustness to spurious maxima point of view. Even if all of them tend to measure, in some way, the same quantity, which is the output independence (or equivalently, the output non-Gaussianity), it is shown that in the case of a mixture involving two sources, the kurtosis-based contrast functions are more robust than the information theoretic ones when the source distributions are multimodal. Index Terms— Blind source separation, contrast function, en- tropy, independent component analysis, kurtosis, multimodal sources. I. INTRODUCTION B LIND SOURCE SEPARATION (BSS) consists in recovering independent source signals from mixtures of them . In this letter, we focus on the linear instantaneous mixture of real sources , where , and denotes the mixing matrix (with a slight abuse of notation, we will omit the temporal variable in the following). At most, one source may have a normal distribution. The mixing system is supposed to be square . Without loss of generality (provided that the sources are stationary and ergodic), it is commonly assumed that the sources are zero-mean and have an identity covariance matrix (i.e., they are sphered ). Most of time, the data are sphered using a prewhitening step: , such that and . If we furthermore constrain the estimated sources (also called “output signals”) to be sphered, they become a rotation trans- form of . If symbolizes the rotation matrix, the mixture scheme can be rewritten as (1) where denotes the transfer matrix between the outputs and the source signals. The aim of BSS is to obtain output signals that Manuscript received July 5, 2004; revised September 7, 2004. The associate editor coordinating the review of this manuscript and approving it for publica- tion was Dr. Yiteng (Arden) Huang. The authors are with the UCL Machine Learning Group, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (e-mail: vrins@ dice.ucl.ac.be). Digital Object Identifier 10.1109/LSP.2004.840863 correspond to the original sources. In this case, the square matrix solution is nonmixing (at most, one nonzero element per row and full rank) [1]; matrix is the rotation matrix, maximizing a so-called contrast function , i.e., . When independent component analysis (ICA) is used to solve the BSS problem, is a function that measures the independence level between the elements of [1]. In order to avoid an exhaustive search in the whole space of orthogonal matrices, a gradient ascent on...
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Ica_spurious_vrins05 - 190 IEEE SIGNAL PROCESSING LETTERS VOL 12 NO 3 MARCH 2005 Information Theoretic versus Cumulant-Based Contrasts for

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