Underdetermined Sparse Blind Source Separation of Nonnegative and Partially Overlapped Data

SIAM Journal on Scientific Computing (Impact Factor: 1.85). 01/2011; 33(4):2063-2094. DOI: 10.1137/100788434
Source: DBLP


We study the solvability of sparse blind separation of $n$ nonnegative sources from $m$ linear mixtures in the underdetermined regime $m<n$. The geometric properties of the mixture matrix and the sparseness structure of the source matrix are closely related to the identification of the mixing matrix. We first illustrate and establish necessary and sufficient conditions for the unique separation for the case of $m$ mixtures and $m+1$ sources, and develop a novel algorithm based on data geometry, source sparseness, and $\ell_1$ minimization. Then we extend the results to any order $m\times n$, $3\leq m<n$, based on the degree of degeneracy of the columns of the mixing matrix. Numerical results substantiate the proposed solvability conditions and show satisfactory performance of our approach.

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Available from: Jack Xin, Oct 16, 2015
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    • "We will also assume that the matrix W is full rank. This is often implicitly assumed in practice otherwise the problem is in general ill-posed, because the matrix H is then typically not uniquely determined; see, e.g., [3] [25]. "
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    ABSTRACT: In this paper, we study the nonnegative matrix factorization problem under the separability assumption (that is, there exists a cone spanned by a small subset of the columns of the input nonnegative data matrix containing all columns), which is equivalent to the hyperspectral unmixing problem under the linear mixing model and the pure-pixel assumption. We present a family of fast recursive algorithms, and prove they are robust under any small perturbations of the input data matrix. This family generalizes several existing hyperspectral unmixing algorithms and hence provides for the first time a theoretical justification of their better practical performance.
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    • "New methods need to be invented to handle this class of data. Recently nonnegative BSS has been attracted considerable attention in NMR spectroscopy [1] [17] [25] [28] [29] [31] [32] [33] [34] [36] [37]. For example, Naanaa and Nuzillard (NN) proposed a nonnegative BSS method in [25] based on a strict local sparseness assumption of the source signals. "
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    Full-text · Article · Oct 2011
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    • "Besides, the properly phased absorption-mode NMR spectral signals from a single-pulse experiment are positive [12]. Recently, there appear considerable activities to nonnegative BSS in NMR spectroscopy with the applications in identification of organic compounds, metabolic fingerprinting, and disease diagnosis [1] [16] [24] [27] [30] [28] [31] [33] [34]. For example, Naanaa and Nuzillard (NN) proposed a nonnegative BSS method in [24] based on a strict local sparseness assumption of the source signals. "
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