The Why and How of Nonnegative Matrix Factorization

Gillis, Nicolas

doi:10.48550/arxiv.1401.5226

Cited by 8 publications

(13 citation statements)

References 83 publications

(123 reference statements)

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“…There are several algorithms available in the literature. Multiplicative update algorithm [103,104], projected gradient method [109], alternating least squares method [29] and several other algorithms described in [15], [27], [65], [94], [95] and [99] are among the algorithms for NMF. The non negative tensor factorizations are described in [56] and several algorithms for both non negative matrix and tensor factorizations with applications can be found in the book [30].…”

Section: Non Negative Matrix Factorization (Nmf)mentioning

confidence: 99%

Literature survey on low rank approximation of matrices

Кумар

Schneider

2016

Linear and Multilinear Algebra

125

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Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic algorithms for low rank approximation. But these techniques are very expensive (O(n 3 ) operations are required for n × n matrices). There are several randomized algorithms available in the literature which are not so expensive as the classical techniques (but the complexity is not linear in n). So, it is very expensive to construct the low rank approximation of a matrix if the dimension of the matrix is very large. There are alternative techniques like Cross/Skeleton approximation which gives the low-rank approximation with linear complexity in n. In this article we review low rank approximation techniques briefly and give extensive references of many techniques.

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Section: Non Negative Matrix Factorization (Nmf)mentioning

confidence: 99%

Literature survey on low rank approximation of matrices

Кумар

Schneider

2016

Linear and Multilinear Algebra

125

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“…It is interesting to note that near-separable NMF algorithms can be used as good initialization strategies for standard NMF algorithms (which usually require some initial guess for U and V ); see the discussion in [17] and the references therein.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation

Gillis¹

2014

SIAM J. Imaging Sci.

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In this paper, we propose a new fast and robust recursive algorithm for near-separable nonnegative matrix factorization, a particular nonnegative blind source separation problem. This algorithm, which we refer to as the successive nonnegative projection algorithm (SNPA), is closely related to the popular successive projection algorithm (SPA), but takes advantage of the nonnegativity constraint in the decomposition. We prove that SNPA is more robust than SPA and can be applied to a broader class of nonnegative matrices. This is illustrated on some synthetic data sets, and on a real-world hyperspectral image.

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“…Hyperspectral unmixing (HU) aims at determining the spectra of the underlying endmembers (or materials) and the corresponding proportions in each sensed pixel from a captured hyperspectral image (HSI). It is an important branch of techniques in hyperspectral data analysis and processing, enables many applications in remote sensing [2], and has close relationship to topics in other contexts, such as non-negative matrix factorization (NMF) in machine learning [3]. Readers are referred to the literature, such as [4,5] and the references therein, for descriptions of various HU approaches.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, a new sparse optimization-based approach was proposed for HU [6]; see also [3,[7][8][9][10]] for other contexts. This approach uses the measured pixel vectors themselves as the (overcomplete) dictionary to perform sparse basis selection.…”

Section: Introductionmentioning

confidence: 99%

“…However, convex SD-MMV relaxations have a drawback-its computational overheads are generally high, since the number of optimization variables involved in a convex SD-MMV relaxation is the square of the number of pixels (e.g., an HSI with 1, 000 pixels amounts to 1, 000, 000 optimization variables). The existing works circumvent this issue by downsizing the HSI data through extra processing [6]; see also [3].…”

Section: Introductionmentioning

confidence: 99%

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Self-Dictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search Are Related

Chan

et al. 2015

IEEE J. Sel. Top. Signal Process.

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This paper considers a recently emerged hyperspectral unmixing formulation based on sparse regression of a self-dictionary multiple measurement vector (SD-MMV) model, wherein the measured hyperspectral pixels are used as the dictionary. Operating under the pure pixel assumption, this SD-MMV formalism is special in that it allows simultaneous identification of the endmember spectral signatures and the number of endmembers. Previous SD-MMV studies mainly focus on convex relaxations. In this study, we explore the alternative of greedy pursuit, which generally provides efficient and simple algorithms. In particular, we design a greedy SD-MMV algorithm using simultaneous orthogonal matching pursuit. Intriguingly, the proposed greedy algorithm is shown to be closely related to some existing pure pixel search algorithms, especially, the successive projection algorithm (SPA). Thus, a link between SD-MMV and pure pixel search is revealed. We then perform exact recovery analyses, and prove that the proposed greedy algorithm is robust to noise-including its identification of the (unknown) number of endmembers-under a sufficiently low noise level. The identification performance of the proposed greedy algorithm is demonstrated through both synthetic and real-data experiments. IntroductionHyperspectral unmixing (HU) aims at determining the spectra of the underlying endmembers (or materials) and the corresponding proportions in each sensed pixel from a captured hyperspectral image (HSI). It is an important branch of techniques in hyperspectral data analysis and processing, * Part of this work was published in EUSIPCO 2013 [1].

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The Why and How of Nonnegative Matrix Factorization

Cited by 8 publications

References 83 publications

Literature survey on low rank approximation of matrices

Literature survey on low rank approximation of matrices

Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation

Self-Dictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search Are Related

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