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Nonnegative factorization and the maximum edge biclique problem
, 2008
"... Nonnegative Matrix Factorization (NMF) is a data analysis technique which allows compression and interpretation of nonnegative data. NMF became widely studied after the publication of the seminal paper by Lee and Seung (Learning the Parts of Objects by Nonnegative Matrix Factorization, Nature, 1999, ..."
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Nonnegative Matrix Factorization (NMF) is a data analysis technique which allows compression and interpretation of nonnegative data. NMF became widely studied after the publication of the seminal paper by Lee and Seung (Learning the Parts of Objects by Nonnegative Matrix Factorization, Nature, 1999, vol. 401, pp. 788–791), which introduced an algorithm based on Multiplicative Updates (MU). More recently, another class of methods called Hierarchical Alternating Least Squares (HALS) was introduced that seems to be much more efficient in practice. In this paper, we consider the problem of approximating a not necessarily nonnegative matrix with the product of two nonnegative matrices, which we refer to as Nonnegative Factorization (NF); this is the subproblem that HALS methods implicitly try to solve at each iteration. We prove that NF is NPhard for any fixed factorization rank, using a reduction to the maximum edge biclique problem. We also generalize the multiplicative updates to NF, which allows us to shed some light on the differences between the MU and HALS algorithms for NMF and give an explanation for the better performance of HALS. Finally, we link stationary points of NF with feasible solutions of the biclique problem to obtain a new type of biclique finding algorithm (based on MU) whose iterations have an algorithmic complexity proportional to the number of edges in the graph, and show that it performs better than comparable existing methods.
Nonnegative Matrix Factorization: A Comprehensive Review
 IEEE TRANS. KNOWLEDGE AND DATA ENG
, 2013
"... Nonnegative Matrix Factorization (NMF), a relatively novel paradigm for dimensionality reduction, has been in the ascendant since its inception. It incorporates the nonnegativity constraint and thus obtains the partsbased representation as well as enhancing the interpretability of the issue corres ..."
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Cited by 17 (2 self)
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Nonnegative Matrix Factorization (NMF), a relatively novel paradigm for dimensionality reduction, has been in the ascendant since its inception. It incorporates the nonnegativity constraint and thus obtains the partsbased representation as well as enhancing the interpretability of the issue correspondingly. This survey paper mainly focuses on the theoretical research into NMF over the last 5 years, where the principles, basic models, properties, and algorithms of NMF along with its various modifications, extensions, and generalizations are summarized systematically. The existing NMF algorithms are divided into four categories: Basic NMF (BNMF),
Dimensionality Reduction, Classification, and Spectral Mixture Analysis using Nonnegative Underapproximation
"... Nonnegative Matrix Factorization (NMF) and its variants have recently been successfully used as dimensionality reduction techniques for identification of the materials present in hyperspectral images. In this paper, we present a new variant of NMF called Nonnegative Matrix Underapproximation (NMU): ..."
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Cited by 8 (7 self)
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Nonnegative Matrix Factorization (NMF) and its variants have recently been successfully used as dimensionality reduction techniques for identification of the materials present in hyperspectral images. In this paper, we present a new variant of NMF called Nonnegative Matrix Underapproximation (NMU): it is based on the introduction of underapproximation constraints which enables one to extract features in a recursive way, like PCA, but preserving nonnegativity. Moreover, we explain why these additional constraints make NMU particularly wellsuited to achieve a partsbased and sparse representation of the data, enabling it to recover the constitutive elements in hyperspectral data. We experimentally show the efficiency of this new strategy on hyperspectral images associated with space object material identification, and on HYDICE and related remote sensing images.
The why and how of nonnegative matrix factorization
 REGULARIZATION, OPTIMIZATION, KERNELS, AND SUPPORT VECTOR MACHINES. CHAPMAN & HALL/CRC
, 2014
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Document Classification Using Nonnegative Matrix Factorization and Underapproximation
 IN PROC. OF THE IEEE INTERNATIONAL SYMPOSIUM ON CIRCUITS AND SYSTEMS (ISCAS), 2009
"... In this study, we use nonnegative matrix factorization (NMF) and nonnegative matrix underapproximation (NMU) approaches to generate feature vectors that can be used to cluster Aviation Safety Reporting System (ASRS) documents obtained from the Distributed National ASAP Archive (DNAA). By preserving ..."
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In this study, we use nonnegative matrix factorization (NMF) and nonnegative matrix underapproximation (NMU) approaches to generate feature vectors that can be used to cluster Aviation Safety Reporting System (ASRS) documents obtained from the Distributed National ASAP Archive (DNAA). By preserving nonnegativity, both the NMF and NMU facilitate a sumofparts representation of the underlying term usage patterns in the ASRS document collection. Both the training and test sets of ASRS documents are parsed and then factored by both algorithms to produce a reducedrank representations of the entire document space. The resulting feature and coefficient matrix factors are used to cluster ASRS documents so that the (known) associated anomalies of training documents are directly mapped to the feature vectors. Dominant features of test documents are then used to generate anomaly relevance scores for those documents. We demonstrate that the approximate solution obtained by NMU using Lagrangrian duality can lead to a better sumofparts representation and document classification accuracy.
Blind Separation of Analytes in Nuclear Magnetic Resonance Spectroscopy and Mass Spectrometry: SparsenessBased Robust Multicomponent Analysis,
 Anal. Chem.
, 2010
"... Abstract We introduce improved model for sparseness constrained nonnegative matrix factorization (sNMF) of amplitude mixtures nuclear magnetic resonance (NMR) spectra into greater number of component spectra. In proposed method selected sNMF algorithm is applied to the square of the amplitude of th ..."
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Abstract We introduce improved model for sparseness constrained nonnegative matrix factorization (sNMF) of amplitude mixtures nuclear magnetic resonance (NMR) spectra into greater number of component spectra. In proposed method selected sNMF algorithm is applied to the square of the amplitude of the mixtures NMR spectra instead to the amplitude spectra itself. Afterwards, the square roots of separated squares of components spectra and concentration matrix yield estimates of the true components amplitude spectra and of concentration matrix. Proposed model 2 remains linear in average when number of overlapping components is increasing, while model based on amplitude spectra of the mixtures moves away from the linear one when number of overlapping components is increased. That is demonstrated through conducted sensitivity analysis. Thus, proposed model improves capability of the sparse NMF algorithms to separate correlated (overlapping) components spectra from smaller number of mixtures NMR spectra. That is demonstrated on two experimental scenarios: extraction of three correlated components spectra from two 1 H NMR mixtures spectra and extraction of four correlated components spectra from three COSY NMR mixtures spectra. Proposed method can increase efficiency in spectral library search by reducing occurrence of false positives and false negatives. That, in turn, can yield better accuracy in biomarker identification studies which makes proposed method important for natural products research and the field of metabolic studies.
Rational Variety Mapping for ContrastEnhanced Nonlinear Unsupervised Segmentation of Multispectral Images of Unstained Specimen
"... A methodology is proposed for nonlinear contrastenhanced unsupervised segmentation of multispectral (color) microscopy images of principally unstained specimens. The methodology exploits spectral diversity and spatial sparseness to find anatomical differences between materials (cells, nuclei, and ba ..."
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A methodology is proposed for nonlinear contrastenhanced unsupervised segmentation of multispectral (color) microscopy images of principally unstained specimens. The methodology exploits spectral diversity and spatial sparseness to find anatomical differences between materials (cells, nuclei, and background) present in the image. It consists of rthorder rational variety mapping (RVM) followed by matrix/tensor factorization. Sparseness constraint implies duality between nonlinear unsupervised segmentation and multiclass pattern assignment problems. Classes not linearly separable in the original input space become separable with high probability in the higherdimensional mapped space. Hence, RVM mapping has two advantages: it takes implicitly
Nonlinear Mixturewise Expansion Approach to Underdetermined Blind Separation of Nonnegative Dependent Sources
"... Abstract Underdetermined blind separation of nonnegative dependent sources consists in decomposing set of observed mixed signals into greater number of original nonnegative and dependent component (source) signals. That is an important problem for which very few algorithms exist. It is also practic ..."
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Abstract Underdetermined blind separation of nonnegative dependent sources consists in decomposing set of observed mixed signals into greater number of original nonnegative and dependent component (source) signals. That is an important problem for which very few algorithms exist. It is also practically relevant for contemporary metabolic profiling of biological samples, such as biomarker identification studies, where sources (a.k.a. pure components or analytes) are aimed to be extracted from mass spectra of complex multicomponent mixtures. This paper presents method for underdetermined blind separation of nonnegative dependent sources. The method performs nonlinear mixturewise mapping of observed data in highdimensional reproducible kernel Hilbert space (RKHS) of functions and sparseness constrained nonnegative matrix factorization (NMF) therein. Thus, original problem is converted into new one with increased number of mixtures, increased number of dependent sources and higherorder (error) terms generated by nonlinear mapping. Provided that amplitudes of original components are sparsely distributed, that is the case for mass spectra of analytes, sparseness constrained NMF in RKHS yields, with significant probability, improved accuracy relative to the case when the same NMF algorithm is performed on original problem. The method is exemplified on numerical and experimental examples related respectively to extraction of ten dependent components from five mixtures and to extraction of ten dependent analytes from mass spectra of two to five mixtures. Thereby, analytes mimic complexity of components expected to be found in biological samples.
NONNEGATIVE MATRIX AND TENSOR FACTORIZATIONS, LEAST SQUARES PROBLEMS, AND APPLICATIONS
, 2011
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Nonlinear Band Expansion and Nonnegative Matrix Underapproximation for Unsupervised Segmentation of a Liver from a Multiphase CT image
"... A methodology is proposed for contrast enhanced unsupervised segmentation of a liver from a twodimensional multiphase CT image. The multiphase CT image is represented by a linear mixture model, whereupon each singlephase CT image is modeled as a linear mixture of spatial distributions of the org ..."
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A methodology is proposed for contrast enhanced unsupervised segmentation of a liver from a twodimensional multiphase CT image. The multiphase CT image is represented by a linear mixture model, whereupon each singlephase CT image is modeled as a linear mixture of spatial distributions of the organs present in the image. The methodology exploits concentration and spatial diversities between organs present in the image and consists of nonlinear dimensionality expansion followed by matrix factorization that relies on sparseness between spatial distributions of organs. Dimensionality expansion increases concentration diversity (contrast) between organs. The methodology is demonstrated on an experimental threephase CT image of a liver of two patients.