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Algorithms for Nonnegative Matrix Factorization
 In NIPS
, 2001
"... Nonnegative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown to minim ..."
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Cited by 1246 (5 self)
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Nonnegative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown
Nonnegative matrix factorization with sparseness constraints,”
 Journal of Machine Learning Research,
, 2004
"... Abstract Nonnegative matrix factorization (NMF) is a recently developed technique for finding partsbased, linear representations of nonnegative data. Although it has successfully been applied in several applications, it does not always result in partsbased representations. In this paper, we sho ..."
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Cited by 498 (0 self)
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Abstract Nonnegative matrix factorization (NMF) is a recently developed technique for finding partsbased, linear representations of nonnegative data. Although it has successfully been applied in several applications, it does not always result in partsbased representations. In this paper, we
Factoring wavelet transforms into lifting steps
 J. FOURIER ANAL. APPL
, 1998
"... This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decompositio ..."
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Cited by 584 (8 self)
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. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We
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 189 (30 self)
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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
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 282 (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
Nonnegative matrix factorization for polyphonic music transcription
 IN PROC. IEEE WORKSHOP APPLICATIONS OF SIGNAL PROCESSING TO AUDIO AND ACOUSTICS
, 2003
"... In this paper we present a methodology for analyzing polyphonic musical passages comprised by notes that exhibit a harmonically fixed spectral profile (such as piano notes). Taking advantage of this unique note structure we can model the audio content of the musical passage by a linear basis transfo ..."
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Cited by 240 (14 self)
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transform and use nonnegative matrix decomposition methods to estimate the spectral profile and the temporal information of every note. This approach results in a very simple and compact system that is not knowledge based, but rather learns notes by observation.
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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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
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
When Does NonNegative Matrix Factorization Give a Correct Decomposition into Parts?
"... We interpret nonnegative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across the faces ..."
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Cited by 200 (1 self)
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We interpret nonnegative matrix factorization geometrically, as the problem of finding a simplicial cone which contains a cloud of data points and which is contained in the positive orthant. We show that under certain conditions, basically requiring that some of the data are spread across
On the equivalence of nonnegative matrix factorization and spectral clustering
 in SIAM International Conference on Data Mining
, 2005
"... Current nonnegative matrix factorization (NMF) deals with X = FG T type. We provide a systematic analysis and extensions of NMF to the symmetric W = HH T, and the weighted W = HSHT. We show that (1) W = HHT is equivalent to Kernel Kmeans clustering and the Laplacianbased spectral clustering. (2) X ..."
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Cited by 159 (20 self)
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Current nonnegative matrix factorization (NMF) deals with X = FG T type. We provide a systematic analysis and extensions of NMF to the symmetric W = HH T, and the weighted W = HSHT. We show that (1) W = HHT is equivalent to Kernel Kmeans clustering and the Laplacianbased spectral clustering. (2
Results 1  10
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