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ON TENSORS, SPARSITY, AND NONNEGATIVE FACTORIZATIONS ∗
"... Abstract. Tensors have found application in a variety of fields, ranging from chemometrics to signal processing and beyond. In this paper, we consider the problem of multilinear modeling of sparse count data. Our goal is to develop a descriptive tensor factorization model of such data, along with ap ..."
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Abstract. Tensors have found application in a variety of fields, ranging from chemometrics to signal processing and beyond. In this paper, we consider the problem of multilinear modeling of sparse count data. Our goal is to develop a descriptive tensor factorization model of such data, along with appropriate algorithms and theory. To do so, we propose that the random variation is best described via a Poisson distribution, which better describes the zeros observed in the data as compared to the typical assumption of a Gaussian distribution. Under a Poisson assumption, we fit a model to observed data using the negative loglikelihood score. We present a new algorithm for Poisson tensor factorization called CANDECOMP–PARAFAC alternating Poisson regression (CPAPR) that is based on a majorizationminimization approach. It can be shown that CPAPR is a generalization of the Lee–Seung multiplicative updates. We show how to prevent the algorithm from converging to nonKKT points and prove convergence of CPAPR under mild conditions. We also explain how to implement CPAPR for largescale sparse tensors and present results on several data sets,bothrealandsimulated.
Utopian: Userdriven topic modeling based on interactive nonnegative matrix factorization
 IEEE TVCG
"... Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/ ..."
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Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/
Activeset Newton algorithm for overcomplete nonnegative representations of audio
 IEEE Transactions on Audio, Speech, and Language Processing
, 2013
"... Abstract—This paper proposes a computationally efficient algorithm for estimating the nonnegative weights of linear combinations of the atoms of largescale audio dictionaries, so that the generalized KullbackLeibler divergence between an audio observation and the model is minimized. This linear m ..."
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Abstract—This paper proposes a computationally efficient algorithm for estimating the nonnegative weights of linear combinations of the atoms of largescale audio dictionaries, so that the generalized KullbackLeibler divergence between an audio observation and the model is minimized. This linear model has been found useful in many audio signal processing tasks, but the existing algorithms are computationally slow when a large number of atoms is used. The proposed algorithm is based on iteratively updating a set of active atoms, with the weights updated using the Newton method and the step size estimated such that the weights remain nonnegative. Algorithm convergence evaluations on representing audio spectra that are mixtures of two speakers show that with all the tested dictionary sizes the proposed method reaches a much lower value of the divergence than can be obtained by conventional algorithms, and is up to 8 times faster. A source separation separation evaluation revealed that when using large dictionaries, the proposed method produces a better separation separation quality in less time. Index Terms—acoustic signal analysis, audio source separation, supervised source separation, nonnegative matrix factorization, Newton algorithm, convex optimization, sparse coding, sparse representation
Nonnegative matrix factorization revisited: Uniqueness and algorithm for symmetric decomposition
 IEEE Trans. Signal Processing
, 2014
"... Abstract—Nonnegative matrix factorization (NMF) has found numerous applications, due to its ability to provide interpretable decompositions. Perhaps surprisingly, existing results regarding its uniqueness properties are rather limited, and there is much room for improvement in terms of algorithms a ..."
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Abstract—Nonnegative matrix factorization (NMF) has found numerous applications, due to its ability to provide interpretable decompositions. Perhaps surprisingly, existing results regarding its uniqueness properties are rather limited, and there is much room for improvement in terms of algorithms as well. Uniqueness aspects of NMF are revisited here from a geometrical point of view. Both symmetric and asymmetric NMF are considered, the former being tantamount to elementwise nonnegative squareroot factorization of positive semidefinite matrices. New uniqueness results are derived, e.g., it is shown that a sufficient condition for uniqueness is that the conic hull of the latent factors is a superset of a particular secondorder cone. Checking this condition is shown to be NPcomplete; yet this and other results offer insights on the role of latent sparsity in this context. On the computational side, a new algorithm for symmetric NMF is proposed, which is very different from existing ones. It alternates between Procrustes rotation and projection onto the nonnegative orthant to find a nonnegative matrix close to the span of the dominant subspace. Simulation results show promising performance with respect to the stateofart. Finally, the new algorithm is applied to a clustering problem for coauthorship data, yielding meaningful and interpretable results. Index Terms—Nonnegative matrix factorization (NMF), symmetric NMF, uniqueness, sparsity, Procrustes rotation.
Fast Nonnegative Tensor Factorization with an ActiveSetLike Method
"... Abstract We introduce an efficient algorithm for computing a lowranknonnegativeCANDECOMP/PARAFAC(NNCP)decomposition.Intextmining, signal processing, and computer vision among other areas, imposing nonnegativity constraints to the lowrank factors of matrices and tensors has been shown an effective ..."
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Abstract We introduce an efficient algorithm for computing a lowranknonnegativeCANDECOMP/PARAFAC(NNCP)decomposition.Intextmining, signal processing, and computer vision among other areas, imposing nonnegativity constraints to the lowrank factors of matrices and tensors has been shown an effective technique providing physically meaningful interpretation. A principled methodology for computing NNCP is alternating nonnegative least squares, in which the nonnegativityconstrained least squares (NNLS) problems are solved in each iteration. In this chapter, we propose to solve the NNLS problems using the block principal pivoting method. The block principal pivoting method overcomes some difficulties of the classical active method for the NNLS problems with a large number of variables. We introducetechniquestoacceleratetheblockprincipalpivotingmethodformultiple righthand sides, which is typical in NNCP computation. Computational experiments show the stateoftheart performance of the proposed method. 1
The why and how of nonnegative matrix factorization
 Regularization, Optimization, Kernels, and Support Vector Machines. Chapman & Hall/CRC
, 2014
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Tripartite graph clustering for dynamic sentiment analysis on social media
, 2014
"... The growing popularity of social media (e.g., Twitter) allows users to easily share information with each other and influence others by expressing their own sentiments on various subjects. In this work, we propose an unsupervised triclustering framework, which analyzes both userlevel and tweetlev ..."
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The growing popularity of social media (e.g., Twitter) allows users to easily share information with each other and influence others by expressing their own sentiments on various subjects. In this work, we propose an unsupervised triclustering framework, which analyzes both userlevel and tweetlevel sentiments through coclustering of a tripartite graph. A compelling feature of the proposed framework is that the quality of sentiment clustering of tweets, users, and features can be mutually improved by joint clustering. We further investigate the evolution of userlevel sentiments and latent feature vectors in an online framework and devise an efficient online algorithm to sequentially update the clustering of tweets, users and features with newly arrived data. The online framework not only provides better quality of both dynamic userlevel and tweetlevel sentiment analysis, but also improves the computational and storage efficiency. We verified the effectiveness and efficiency of the proposed approaches on the November 2012 California ballot Twitter data. 1.
Hierarchical Clustering of Hyperspectral Images Using RankTwo Nonnegative Matrix Factorization
 IEEE, Transactions on Geoscience and Remote Sensing
, 2015
"... In this paper, we design a hierarchical clustering algorithm for highresolution hyperspectral images. At the core of the algorithm, a new ranktwo nonnegative matrix factorizations (NMF) algorithm is used to split the clusters, which is motivated by convex geometry concepts. The method starts with ..."
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In this paper, we design a hierarchical clustering algorithm for highresolution hyperspectral images. At the core of the algorithm, a new ranktwo nonnegative matrix factorizations (NMF) algorithm is used to split the clusters, which is motivated by convex geometry concepts. The method starts with a single cluster containing all pixels, and, at each step, (i) selects a cluster in such a way that the error at the next step is minimized, and (ii) splits the selected cluster into two disjoint clusters using ranktwo NMF in such a way that the clusters are well balanced and stable. The proposed method can also be used as an endmember extraction algorithm in the presence of pure pixels. The effectiveness of this approach is illustrated on several synthetic and realworld hyperspectral images, and shown to outperform standard clustering techniques such as kmeans, spherical kmeans and standard NMF.
Fast Bregman Divergence NMF using Taylor Expansion and Coordinate Descent
"... Nonnegative matrix factorization (NMF) provides a lower rank approximation of a matrix. Due to nonnegativity imposed on the factors, it gives a latent structure that is often more physically meaningful than other lower rank approximations such as singular value decomposition (SVD). Most of the algo ..."
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Nonnegative matrix factorization (NMF) provides a lower rank approximation of a matrix. Due to nonnegativity imposed on the factors, it gives a latent structure that is often more physically meaningful than other lower rank approximations such as singular value decomposition (SVD). Most of the algorithms proposed in literature for NMF have been based on minimizing the Frobenius norm. This is partly due to the fact that the minimization problem based on the Frobenius norm provides much more flexibility in algebraic manipulation than other divergences. In this paper we propose a fast NMF algorithm that is applicable to general Bregman divergences. Through Taylor series expansion of the Bregman divergences, we reveal a relationship between Bregman divergences and Euclidean distance. This key relationship provides a new direction for NMF algorithms with general Bregman divergences when combined with the scalar block coordinate descent method. The proposed algorithm generalizes several recently proposed methods for computation of NMF with Bregman divergences and is computationally faster than existing alternatives. We demonstrate the effectiveness of our approach with experiments conducted on artificial as well as real world data.
Visual Steering and Verification of Mass Spectrometry Data Factorization in Air Quality Research
"... Abstract — Thestudyofaerosolcompositionforairqualityresearchinvolves the analysis of highdimensional single particle mass spectrometry data. We describe, apply, and evaluate a novel interactive visual framework for dimensionality reduction ofsuchdata. Our framework is based on nonnegative matrix f ..."
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Abstract — Thestudyofaerosolcompositionforairqualityresearchinvolves the analysis of highdimensional single particle mass spectrometry data. We describe, apply, and evaluate a novel interactive visual framework for dimensionality reduction ofsuchdata. Our framework is based on nonnegative matrix factorization with specifically defined regularization terms that aid in resolving mass spectrum ambiguity. Thereby, visualization assumes a key role in providing insight into and allowing to actively control a heretofore elusive data processing step, and thus enabling rapid analysis meaningful to domain scientists. In extending existing black box schemes, we explore design choices for visualizing, interacting with, and steering the factorization process to produce physically meaningful results. A domainexpert evaluation of our system performed bytheairqualityresearchexpertsinvolvedinthisefforthas shown that our method and prototype admits the finding of unambiguous and physicallycorrectlowerdimensionalbasistransformations of mass spectrometry data at significantly increased speed and a higher degree of ease. Index Terms—Dimension reduction, mass spectrometry data, validation and verification of matrix factorization, visual encodings of numerical error metrics, multidimensional data visualization. 1