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Blind Source Separation in Nonlinear Mixtures

Abstract : Blind Source Separation (BSS) is a technique for estimating individual source components from their mixtures at multiple sensors, where the mixing model is unknown. Although it has been mathematically shown that for linear mixtures, under mild conditions, mutually independent sources can be reconstructed up to accepted ambiguities, there is not such theoretical basis for general nonlinear models. This is why there are relatively few results in the literature in this regard in the recent decades, which are focused on specific structured nonlinearities.In the present study, the problem is tackled using a novel approach utilizing temporal information of the signals. The original idea followed in this purpose is to study a linear time-varying source separation problem deduced from the initial nonlinear problem by derivations. It is shown that already-proposed counter-examples showing inefficiency of Independent Component Analysis (ICA) for nonlinear mixtures, loose their validity, considering independence in the sense of stochastic processes instead of simple random variables. Based on this approach, both nice theoretical results and algorithmic developments are provided. Even though these achievements are not claimed to be a mathematical proof for the separability of nonlinear mixtures, it is shown that given a few assumptions, which are satisfied in most practical applications, they are separable.Moreover, nonlinear BSS for two useful sets of source signals is also addressed: (1) spatially sparse sources and (2) Gaussian processes. Distinct BSS methods are proposed for these two cases, each of which has been widely studied in the literature and has been shown to be quite beneficial in modeling many practical applications.Concerning Gaussian processes, it is demonstrated that not all nonlinear mappings can preserve Gaussianity of the input. For example being restricted to polynomial functions, the only Gaussianity-preserving function is linear. This idea is utilized for proposing a linearizing algorithm which, cascaded by a conventional linear BSS method, separates polynomial mixturesof Gaussian processes.Concerning spatially sparse sources, it is shown that spatially sparsesources make manifolds in the observations space, and can be separated once the manifolds are clustered and learned. For this purpose, multiple manifold learning problem has been generally studied, whose results are not limited to the proposed BSS framework and can be employed in other topics requiring a similar issue.
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Bahram Ehsandoust. Blind Source Separation in Nonlinear Mixtures. Signal and Image Processing. Sharif University of Technology (Tehran), 2018. English. ⟨NNT : 2018GREAT033⟩. ⟨tel-01885816⟩

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