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Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers
, 2010
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Nonnegative matrix factorization with sparseness constraints
 Jour. of
, 2004
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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 317 (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.
Projected gradient methods for Nonnegative Matrix Factorization
 Neural Computation
, 2007
"... Nonnegative matrix factorization (NMF) can be formulated as a minimization problem with bound constraints. Although boundconstrained optimization has been studied extensively in both theory and practice, so far no study has formally applied its techniques to NMF. In this paper, we propose two proj ..."
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Cited by 275 (2 self)
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Nonnegative matrix factorization (NMF) can be formulated as a minimization problem with bound constraints. Although boundconstrained optimization has been studied extensively in both theory and practice, so far no study has formally applied its techniques to NMF. In this paper, we propose two projected gradient methods for NMF, both of which exhibit strong optimization properties. We discuss efficient implementations and demonstrate that one of the proposed methods converges faster than the popular multiplicative update approach. A simple MATLAB code is also provided. 1
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 199 (7 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.
Learning Spatially Localized, PartsBased Representation
, 2001
"... In this paper, we propose a novel method, called local nonnegative matrix factorization (LNMF), for learning spatially localized, partsbased subspace representation of visual patterns. An objective function is defined to impose localization constraint, in addition to the nonnegativity constraint i ..."
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Cited by 196 (7 self)
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In this paper, we propose a novel method, called local nonnegative matrix factorization (LNMF), for learning spatially localized, partsbased subspace representation of visual patterns. An objective function is defined to impose localization constraint, in addition to the nonnegativity constraint in the standard NMF [1]. This gives a set of bases which not only allows a nonsubtractive (partbased) representation of images but also manifests localized features. An algorithm is presented for the learning of such basis components. Experimental results are presented to compare LNMF with the NMF and PCA methods for face representation and recognition, which demonstrates advantages of LNMF.
Monaural sound source separation by nonnegative matrix factorization with temporal continuity and sparseness criteria
 IEEE Trans. On Audio, Speech and Lang. Processing
, 2007
"... Abstract—An unsupervised learning algorithm for the separation of sound sources in onechannel music signals is presented. The algorithm is based on factorizing the magnitude spectrogram of an input signal into a sum of components, each of which has a fixed magnitude spectrum and a timevarying gain ..."
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Cited by 185 (30 self)
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Abstract—An unsupervised learning algorithm for the separation of sound sources in onechannel music signals is presented. The algorithm is based on factorizing the magnitude spectrogram of an input signal into a sum of components, each of which has a fixed magnitude spectrum and a timevarying gain. Each sound source, in turn, is modeled as a sum of one or more components. The parameters of the components are estimated by minimizing the reconstruction error between the input spectrogram and the model, while restricting the component spectrograms to be nonnegative and favoring components whose gains are slowly varying and sparse. Temporal continuity is favored by using a cost term which is the sum of squared differences between the gains in adjacent frames, and sparseness is favored by penalizing nonzero gains. The proposed iterative estimation algorithm is initialized with random values, and the gains and the spectra are then alternatively updated using multiplicative update rules until the values converge. Simulation experiments were carried out using generated mixtures of pitched musical instrument samples and drum sounds. The performance of the proposed method was compared with independent subspace analysis and basic nonnegative matrix factorization, which are based on the same linear model. According to these simulations, the proposed method enables a better separation quality than the previous algorithms. Especially, the temporal continuity criterion improved the detection of pitched musical sounds. The sparseness criterion did not produce significant improvements. Index Terms—Acoustic signal analysis, audio source separation, blind source separation, music, nonnegative matrix factorization, sparse coding, unsupervised learning. I.
Nonnegative sparse coding, in
 Proc. IEEE Workshop on Neural Networks for Signal Processing (NNSP’2002), 2002
"... Abstract. Nonnegative sparse coding is a method for decomposing multivariate data into nonnegative sparse components. In this paper we briefly describe the motivation behind this type of data representation and its relation to standard sparse coding and nonnegative matrix factorization. We then gi ..."
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Cited by 167 (3 self)
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Abstract. Nonnegative sparse coding is a method for decomposing multivariate data into nonnegative sparse components. In this paper we briefly describe the motivation behind this type of data representation and its relation to standard sparse coding and nonnegative matrix factorization. We then give a simple yet efficient multiplicative algorithm for finding the optimal values of the hidden components. In addition, we show how the basis vectors can be learned from the observed data. Simulations demonstrate the effectiveness of the proposed method.
A generalization of principal component analysis to the exponential family
 Advances in Neural Information Processing Systems
, 2001
"... Principal component analysis (PCA) is a commonly applied technique for dimensionality reduction. PCA implicitly minimizes a squared loss function, which may be inappropriate for data that is not realvalued, such as binaryvalued data. This paper draws on ideas from the Exponential family, Generaliz ..."
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Cited by 154 (1 self)
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Principal component analysis (PCA) is a commonly applied technique for dimensionality reduction. PCA implicitly minimizes a squared loss function, which may be inappropriate for data that is not realvalued, such as binaryvalued data. This paper draws on ideas from the Exponential family, Generalized linear models, and Bregman distances, to give a generalization of PCA to loss functions that we argue are better suited to other data types. We describe algorithms for minimizing the loss functions, and give examples on simulated data. 1