# Algebraic identification of under-determined mixtures

Abstract : We consider the linear statistical model x=As+b, where x denotes the P-dimensional vector of observations, s the N-dimensional source vector, A the PxN mixing matrix and b an additive noise, which stands for background noise as well as modeling errors. Matrix A is unknown deterministic, whereas s and b are random and also unobserved. All quantities involved are assumed to take their values in the real or complex ﬁeld. It is assumed that components sn of vector s are statistically mutually independent, and that random vectors b and s are statistically independent. The particularity of this chapter is that the number of sources, N, is assumed to be strictly larger than the number of sensors, P. Even if the mixing matrix were known, it would in general be quite difficult to recover the sources. In fact the mixing matrix does not admit a left inverse, because the linear system is under-determined, which means that it has more unknowns than equations. The goal is to identify the mixing matrix A from the sole observation of realizations of vector x. Note that other approaches exist that do not assume statistical independence among sources. One can mention non negativity of sources and mixture (see Chapter 13), ﬁnite alphabet (see Chapters 6 and 12) with possibly a sparsity assumption on source values (see Chapter 10). This chapter is organized as follows. General assumptions are stated in Section 9.1. Necessary conditions under which the identiﬁcation problem is well posed are pointed out in Section 9.2. Various ways of posing the problem in mathematical terms are described in Section 9.3. Then tensor tools are introduced in Section 9.4 in order to describe numerical algorithms in Section 9.5.
Keywords :
Type de document :
Chapitre d'ouvrage
P. Comon and C. Jutten. Handbook of Blind Source Separation, Academic Press (Elsevier), pp.325-365, 2010
Domaine :

https://hal.archives-ouvertes.fr/hal-00482242
Contributeur : Pierre Comon <>
Soumis le : lundi 10 mai 2010 - 09:45:56
Dernière modification le : mardi 8 juin 2010 - 16:22:16

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• HAL Id : hal-00482242, version 1

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Pierre Comon, Lieven Delathauwer. Algebraic identification of under-determined mixtures. P. Comon and C. Jutten. Handbook of Blind Source Separation, Academic Press (Elsevier), pp.325-365, 2010. <hal-00482242>

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