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

ABSTRACT 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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    • "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.
    IEEE Transactions on Pattern Analysis and Machine Intelligence 08/2012; 36(4). DOI:10.1109/TPAMI.2013.226 · 5.78 Impact Factor
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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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    ABSTRACT: In this paper, we develop a novel blind source separation (BSS) method for nonnegative and correlated data, particularly for the nearly degenerate data. The motivation lies in nuclear magnetic resonance (NMR) spectroscopy, where a multiple mixture NMR spectra are recorded to identify chemical compounds with similar structures (degeneracy). There have been a number of successful approaches for solving BSS problems by exploiting the nature of source signals. For instance, independent component analysis (ICA) is used to separate statistically independent (orthogonal) source signals. However, signal orthogonality is not guaranteed in many real-world problems. This new BSS method developed here deals with nonorthogonal signals. The independence assumption is replaced by a condition which requires dominant interval(s) (DI) from each of source signals over others. Additionally, the mixing matrix is assumed to be nearly singular. The method first estimates the mixing matrix by exploiting geometry in data clustering. Due to the degeneracy of the data, a small deviation in the estimation may introduce errors (spurious peaks of negative values in most cases) in the output. To resolve this challenging problem and improve robustness of the separation, methods are developed in two aspects. One technique is to find a better estimation of the mixing matrix by allowing a constrained perturbation to the clustering output, and it can be achieved by a quadratic programming. The other is to seek sparse source signals by exploiting the DI condition, and it solves an $\ell_1$ optimization. We present numerical results of NMR data to show the performance and reliability of the method in the applications arising in NMR spectroscopy.
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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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    ABSTRACT: Motivated by the nuclear magnetic resonance (NMR) spectroscopy of biofluids (urine and blood serum), we present a recursive blind source separation (rBSS) method for nonnegative and correlated data. BSS problem arises when one attempts to recover a set of source signals from a set of mixture signals without knowing the mixing process. Various approaches have been developed to solve BSS problems relying on the assumption of statistical independence of the source signals. However, signal independence is not guaranteed in many real-world data like the NMR spectra of chemical compounds. The rBSS method introduced in this paper deals with the nonnegative and correlated signals arising in NMR spectroscopy of biofluids. The statistical independence requirement is replaced by a constraint which requires dominant interval(s) from each source signal over some of the other source signals in a hierarchical manner. This condition is applicable for many real-world signals such as NMR spectra of urine and blood serum for metabolic fingerprinting and disease diagnosis. Exploiting the hierarchically dominant intervals from the source signals, the rBSS method reduces the BSS problem into a series of sub-BSS problems by a combination of data clustering, linear programming, and successive elimination of variables. Then in each sub-BSS problem, an ℓ 1 minimization problem is formulated for recovering the source signals in a sparse transformed domain. The method is substantiated by examples from NMR spectroscopy data and is promising towards separation and detection in complex chemical spectra without the expensive multi-dimensional NMR data. KeywordsNonnegative and correlated sources–Blind source separation–Recursive method–Data clustering– ℓ 1 minimization
    Journal of Scientific Computing 06/2011; 51(3):1-21. DOI:10.1007/s10915-011-9528-9 · 1.70 Impact Factor
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