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282
Online learning for matrix factorization and sparse coding
, 2010
"... Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the largescale matrix factorization problem that consists of learning the basis set in order to ad ..."
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Cited by 330 (31 self)
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Sparse coding—that is, modelling data vectors as sparse linear combinations of basis elements—is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the largescale matrix factorization problem that consists of learning the basis set in order to adapt it to specific data. Variations of this problem include dictionary learning in signal processing, nonnegative matrix factorization and sparse principal component analysis. In this paper, we propose to address these tasks with a new online optimization algorithm, based on stochastic approximations, which scales up gracefully to large data sets with millions of training samples, and extends naturally to various matrix factorization formulations, making it suitable for a wide range of learning problems. A proof of convergence is presented, along with experiments with natural images and genomic data demonstrating that it leads to stateoftheart performance in terms of speed and optimization for both small and large data sets.
Algorithms and applications for approximate nonnegative matrix factorization
 Computational Statistics and Data Analysis
, 2006
"... In this paper we discuss the development and use of lowrank approximate nonnegative matrix factorization (NMF) algorithms for feature extraction and identification in the fields of text mining and spectral data analysis. The evolution and convergence properties of hybrid methods based on both spars ..."
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Cited by 204 (8 self)
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In this paper we discuss the development and use of lowrank approximate nonnegative matrix factorization (NMF) algorithms for feature extraction and identification in the fields of text mining and spectral data analysis. The evolution and convergence properties of hybrid methods based on both sparsity and smoothness constraints for the resulting nonnegative matrix factors are discussed. The interpretability of NMF outputs in specific contexts are provided along with opportunities for future work in the modification of NMF algorithms for largescale and timevarying datasets. Key words: nonnegative matrix factorization, text mining, spectral data analysis, email surveillance, conjugate gradient, constrained least squares.
Sparse nonnegative matrix factorizations via alternating nonnegativityconstrained least squares for microarray data analysis
 VOL. 23 NO. 12 2007, PAGES 1495–1502 BIOINFORMATICS
, 2007
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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
Reidentification by relative distance comparison
 In PAMI
, 2013
"... Abstract—Matching people across nonoverlapping camera views at different locations and different times, known as person reidentification, is both a hard and important problem for associating behavior of people observed in a large distributed space over a prolonged period of time. Person reidentifica ..."
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Cited by 55 (8 self)
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Abstract—Matching people across nonoverlapping camera views at different locations and different times, known as person reidentification, is both a hard and important problem for associating behavior of people observed in a large distributed space over a prolonged period of time. Person reidentification is fundamentally challenging because of the large visual appearance changes caused by variations in view angle, lighting, background clutter, and occlusion. To address these challenges, most previous approaches aim to model and extract distinctive and reliable visual features. However, seeking an optimal and robust similarity measure that quantifies a wide range of features against realistic viewing conditions from a distance is still an open and unsolved problem for person reidentification. In this paper, we formulate person reidentification as a relative distance comparison (RDC) learning problem in order to learn the optimal similarity measure between a pair of person images. This approach avoids treating all features indiscriminately and does not assume the existence of some universally distinctive and reliable features. To that end, a novel relative distance comparison model is introduced. The model is formulated to maximize the likelihood of a pair of true matches having a relatively smaller distance than that of a wrong match pair in a soft discriminant manner. Moreover, in order to maintain the tractability of the model in large scale learning, we further develop an ensemble RDC model. Extensive experiments on three publicly available benchmarking datasets are carried out to demonstrate the clear superiority of the proposed RDC models over related popular person reidentification techniques. The results also show that the new RDC models are more robust against visual appearance changes and less susceptible to model overfitting compared to other related existing models. Index Terms—Person reidentification, feature quantification, feature selection, relative distance comparison Ç 1
On the Convergence of Multiplicative Update Algorithms for Nonnegative Matrix Factorization
"... Nonnegative matrix factorization (NMF) is useful to find basis information of nonnegative data. Currently multiplicative updates are a simple and popular way to find the factorization. However, there is no proofs showing that such updates converge to a stationary point of the NMF optimization prob ..."
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Cited by 54 (0 self)
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Nonnegative matrix factorization (NMF) is useful to find basis information of nonnegative data. Currently multiplicative updates are a simple and popular way to find the factorization. However, there is no proofs showing that such updates converge to a stationary point of the NMF optimization problem. Stationarity is important as it is a necessary condition of a local minimum. This paper discusses the difficulty of proving the convergence. We propose slight modifications of existing updates and prove their convergence. Techniques invented in this paper can be applied to prove the convergence for other boundconstrained optimization problems. 1
A Simple Algorithm for Nuclear Norm Regularized Problems
"... Optimization problems with a nuclear norm regularization, such as e.g. low norm matrix factorizations, have seen many applications recently. We propose a new approximation algorithm building upon the recent sparse approximate SDP solver of (Hazan, 2008). The experimental efficiency of our method is ..."
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Cited by 49 (3 self)
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Optimization problems with a nuclear norm regularization, such as e.g. low norm matrix factorizations, have seen many applications recently. We propose a new approximation algorithm building upon the recent sparse approximate SDP solver of (Hazan, 2008). The experimental efficiency of our method is demonstrated on large matrix completion problems such as the Netflix dataset. The algorithm comes with strong convergence guarantees, and can be interpreted as a first theoretically justified variant of SimonFunktype SVD heuristics. The method is free of tuning parameters, and very easy to parallelize. 1.
Fast Local Algorithms for Large Scale Nonnegative Matrix and Tensor Factorizations
, 2008
"... Nonnegative matrix factorization (NMF) and its extensions such as Nonnegative Tensor Factorization (NTF) have become prominent techniques for blind sources separation (BSS), analysis of image databases, data mining and other information retrieval and clustering applications. In this paper we propose ..."
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Cited by 49 (13 self)
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Nonnegative matrix factorization (NMF) and its extensions such as Nonnegative Tensor Factorization (NTF) have become prominent techniques for blind sources separation (BSS), analysis of image databases, data mining and other information retrieval and clustering applications. In this paper we propose a family of efficient algorithms for NMF/NTF, as well as sparse nonnegative coding and representation, that has many potential applications in computational neuroscience, multisensory processing, compressed sensing and multidimensional data analysis. We have developed a class of optimized local algorithms which are referred to as Hierarchical Alternating Least Squares (HALS) algorithms. For these purposes, we have performed sequential constrained minimization on a set of squared Euclidean distances. We then extend this approach to robust cost functions using the Alpha and Beta divergences and derive flexible update rules. Our algorithms are locally stable and work well for NMFbased blind source separation (BSS) not only for the overdetermined case but also for an underdetermined (overcomplete) case (i.e., for a system which has less sensors than sources) if data are sufficiently sparse. The NMF learning rules are extended and generalized for Nth order nonnegative tensor factorization (NTF). Moreover, these algorithms can be tuned to different noise statistics by adjusting a single parameter. Extensive experimental results confirm the accuracy and computational performance of the developed algorithms, especially, with usage of multilayer hierarchical NMF approach [3].
Fast newtontype methods for the least squares nonnegative matrix approximation problem
 Statistical Analysis and Data Mining
, 2008
"... Nonnegative Matrix Approximation is an effective matrix decomposition technique that has proven to be useful for a wide variety of applications ranging from document analysis and image processing to bioinformatics. There exist a few algorithms for nonnegative matrix approximation (NNMA), for example ..."
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Cited by 45 (6 self)
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Nonnegative Matrix Approximation is an effective matrix decomposition technique that has proven to be useful for a wide variety of applications ranging from document analysis and image processing to bioinformatics. There exist a few algorithms for nonnegative matrix approximation (NNMA), for example, Lee & Seung’s multiplicative updates, alternating least squares, and certain gradient descent based procedures. All of these procedures suffer from either slow convergence, numerical instabilities, or at worst, theoretical unsoundness. In this paper we present new and improved algorithms for the leastsquares NNMA problem, which are not only theoretically wellfounded, but also overcome many of the deficiencies of other methods. In particular, we use nondiagonal gradient scaling to obtain rapid convergence. Our methods provide numerical results superior to both Lee & Seung’s method as well to the alternating least squares (ALS) heuristic, which is known to work well in some situations but has no theoretical guarantees (Berry et al. 2006). Our approach extends naturally to include regularization and boxconstraints, without sacrificing convergence guarantees. We present experimental results on both synthetic and realworld datasets to demonstrate the superiority of our methods, in terms of better approximations as well as efficiency.