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Convex and SemiNonnegative Matrix Factorizations
, 2008
"... We present several new variations on the theme of nonnegative matrix factorization (NMF). Considering factorizations of the form X = F GT, we focus on algorithms in which G is restricted to contain nonnegative entries, but allow the data matrix X to have mixed signs, thus extending the applicable ra ..."
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Cited by 112 (10 self)
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We present several new variations on the theme of nonnegative matrix factorization (NMF). Considering factorizations of the form X = F GT, we focus on algorithms in which G is restricted to contain nonnegative entries, but allow the data matrix X to have mixed signs, thus extending the applicable range of NMF methods. We also consider algorithms in which the basis vectors of F are constrained to be convex combinations of the data points. This is used for a kernel extension of NMF. We provide algorithms for computing these new factorizations and we provide supporting theoretical analysis. We also analyze the relationships between our algorithms and clustering algorithms, and consider the implications for sparseness of solutions. Finally, we present experimental results that explore the properties of these new methods.
NONNEGATIVE MATRIX FACTORIZATION BASED ON ALTERNATING NONNEGATIVITY CONSTRAINED LEAST SQUARES AND ACTIVE SET METHOD
"... The nonnegative matrix factorization (NMF) determines a lower rank approximation of a ¢¤£¦¥¨§�©���� �� � matrix where an ������������������ � interger is given and nonnegativity is imposed on all components of the factors applied to numerous data analysis problems. In applications where the compone ..."
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Cited by 86 (7 self)
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The nonnegative matrix factorization (NMF) determines a lower rank approximation of a ¢¤£¦¥¨§�©���� �� � matrix where an ������������������ � interger is given and nonnegativity is imposed on all components of the factors applied to numerous data analysis problems. In applications where the components of the data are necessarily nonnegative such as chemical concentrations in experimental results or pixels in digital images, the NMF provides a more relevant interpretation of the results since it gives nonsubtractive combinations of nonnegative basis vectors. In this paper, we introduce an algorithm for the NMF based on alternating nonnegativity constrained least squares (NMF/ANLS) and the active set based fast algorithm for nonnegativity constrained least squares with multiple right hand side vectors, and discuss its convergence properties and a rigorous convergence criterion based on the KarushKuhnTucker (KKT) conditions. In addition, we also describe algorithms for sparse NMFs and regularized NMF. We show how we impose a sparsity constraint on one of the factors by �� �norm minimization and discuss its convergence properties. Our algorithms are compared to other commonly used NMF algorithms in the literature on several test data sets in terms of their convergence behavior. £�¥�§�©� � and � £�¥���©� �. The NMF has attracted much attention for over a decade and has been successfully
Toward Faster Nonnegative Matrix Factorization: A New Algorithm and Comparisons
"... Nonnegative Matrix Factorization (NMF) is a dimension reduction method that has been widely used for various tasks including text mining, pattern analysis, clustering, and cancer class discovery. The mathematical formulation for NMF appears as a nonconvex optimization problem, and various types of ..."
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Cited by 40 (5 self)
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Nonnegative Matrix Factorization (NMF) is a dimension reduction method that has been widely used for various tasks including text mining, pattern analysis, clustering, and cancer class discovery. The mathematical formulation for NMF appears as a nonconvex optimization problem, and various types of algorithms have been devised to solve the problem. The alternating nonnegative least squares (ANLS) framework is a block coordinate descent approach for solving NMF, which was recently shown to be theoretically sound and empirically efficient. In this paper, we present a novel algorithm for NMF based on the ANLS framework. Our new algorithm builds upon the block principal pivoting method for the nonnegativity constrained least squares problem that overcomes some limitations of active set methods. We introduce ideas to efficiently extend the block principal pivoting method within the context of NMF computation. Our algorithm inherits the convergence theory of the ANLS framework and can easily be extended to other constrained NMF formulations. Comparisons of algorithms using datasets that are from real life applications as well as those artificially generated show that the proposed new algorithm outperforms existing ones in computational speed. 1
Fast nonnegative matrix factorization: An activesetlike method and comparisons
 SIAM Journal on Scientific Computing
, 2011
"... Abstract. Nonnegative matrix factorization (NMF) is a dimension reduction method that has been widelyused fornumerousapplications including text mining, computer vision, pattern discovery, and bioinformatics. A mathematical formulation for NMF appears as a nonconvex optimization problem, and variou ..."
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Cited by 35 (6 self)
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Abstract. Nonnegative matrix factorization (NMF) is a dimension reduction method that has been widelyused fornumerousapplications including text mining, computer vision, pattern discovery, and bioinformatics. A mathematical formulation for NMF appears as a nonconvex optimization problem, and various types of algorithms have been devised to solve the problem. The alternating nonnegative leastsquares (ANLS)frameworkisablock coordinate descent approach forsolving NMF, which was recently shown to be theoretically sound and empiricallyefficient. In this paper, we present a novel algorithm for NMF based on the ANLS framework. Our new algorithm builds upon the block principal pivoting method for the nonnegativityconstrained least squares problem that overcomes a limitation of the active set method. We introduce ideas that efficiently extend the block principal pivoting method within the context of NMF computation. Our algorithm inherits the convergence property of the ANLS framework and can easily be extended to other constrained NMF formulations. Extensive computational comparisons using data sets that are from real life applications as well as those artificially generated show that the proposed algorithm provides stateoftheart performance in terms of computational speed.
Symmetric Nonnegative Matrix Factorization for Graph Clustering
"... Nonnegative matrix factorization (NMF) provides a lower rank approximation of a nonnegative matrix, and has been successfully used as a clustering method. In this paper, we offer some conceptual understanding for the capabilities and shortcomings of NMF as a clustering method. Then, we propose Symme ..."
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Cited by 21 (4 self)
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Nonnegative matrix factorization (NMF) provides a lower rank approximation of a nonnegative matrix, and has been successfully used as a clustering method. In this paper, we offer some conceptual understanding for the capabilities and shortcomings of NMF as a clustering method. Then, we propose Symmetric NMF (SymNMF) as a general framework for graph clustering, which inherits the advantages of NMF by enforcing nonnegativity on the clustering assignment matrix. Unlike NMF, however, SymNMF is based on a similarity measure between data points, and factorizes a symmetric matrix containing pairwise similarity values (not necessarily nonnegative). We compare SymNMF with the widelyused spectral clustering methods, and give an intuitive explanation of why SymNMF captures the cluster structure embedded in the graph representation more naturally. In addition, we develop a Newtonlike algorithm that exploits secondorder information efficiently, so as to show the feasibility of SymNMF as a practical framework for graph clustering. Our experiments on artificial graph data, text data, and image data demonstrate the substantially enhanced clustering quality of SymNMF over spectral clustering and NMF. Therefore, SymNMF is able to achieve better clustering results on both linear and nonlinear manifolds, and serves as a potential basis for many extensions and applications. 1
Linear and Nonlinear Projective Nonnegative Matrix Factorization
"... Abstract—A variant of nonnegative matrix factorization (NMF) which was proposed earlier is analyzed here. It is called Projective Nonnegative Matrix Factorization (PNMF). The new method approximately factorizes a projection matrix, minimizing the reconstruction error, into a positive lowrank matrix ..."
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Cited by 21 (3 self)
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Abstract—A variant of nonnegative matrix factorization (NMF) which was proposed earlier is analyzed here. It is called Projective Nonnegative Matrix Factorization (PNMF). The new method approximately factorizes a projection matrix, minimizing the reconstruction error, into a positive lowrank matrix and its transpose. The dissimilarity between the original data matrix and its approximation can be measured by the Frobenius matrix norm or the modified KullbackLeibler divergence. Both measures are minimized by multiplicative update rules, whose convergence is proven for the first time. Enforcing orthonormality to the basic objective is shown to lead to an even more efficient update rule, which is also readily extended to nonlinear cases. The formulation of the PNMF objective is shown to be connected to a variety of existing nonnegative matrix factorization methods and clustering approaches. In addition, the derivation using Lagrangian multipliers reveals the relation between reconstruction and sparseness. For kernel principal component analysis with the binary constraint, useful in graph partitioning problems, the nonlinear kernel PNMF provides a good approximation which outperforms an existing discretization approach. Empirical study on three realworld databases shows that PNMF can achieve the best or close to the best in clustering. The proposed algorithm runs more efficiently than the compared nonnegative matrix factorization methods, especially for highdimensional data. Moreover, contrary to the basic NMF, the trained projection matrix can be readily used for newly coming samples and demonstrates good generalization. I.
Nonnegativity Constraints in Numerical Analysis
"... A survey of the development of algorithms for enforcing nonnegativity constraints in scientific computation is given. Special emphasis is placed on such constraints in least squares computations in numerical linear algebra and in nonlinear optimization. Techniques involving nonnegative lowrank matr ..."
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Cited by 20 (2 self)
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A survey of the development of algorithms for enforcing nonnegativity constraints in scientific computation is given. Special emphasis is placed on such constraints in least squares computations in numerical linear algebra and in nonlinear optimization. Techniques involving nonnegative lowrank matrix and tensor factorizations are also emphasized. Details are provided for some important classical and modern applications in science and engineering. For completeness, this report also includes an effort toward a literature survey of the various algorithms and applications of nonnegativity constraints in numerical analysis. Key Words: nonnegativity constraints, nonnegative least squares, matrix and tensor factorizations, image processing, optimization.
Nonnegative matrix factorization of partial track data for motion segmentation
 In International Conference on Computer Vision (ICCV
, 2009
"... This paper addresses the problem of segmenting lowlevel partial feature point tracks belonging to multiple motions. We show that the local velocity vectors at each instant of the trajectory are an effective basis for motion segmentation. We decompose the velocity profiles of point tracks into dif ..."
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Cited by 19 (0 self)
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This paper addresses the problem of segmenting lowlevel partial feature point tracks belonging to multiple motions. We show that the local velocity vectors at each instant of the trajectory are an effective basis for motion segmentation. We decompose the velocity profiles of point tracks into different motion components and corresponding nonnegative weights using nonnegative matrix factorization (NNMF). We then segment the different motions using spectral clustering on the derived weights. We test our algorithm on the Hopkins 155 benchmarking database and several new sequences, demonstrating that the proposed algorithm can accurately segment multiple motions at a speed of a few seconds per frame. We show that our algorithm is particularly successful on lowlevel tracks from realworld video that are fragmented, noisy and inaccurate. 1.
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),