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28
Tensor Decompositions and Applications
 SIAM REVIEW
, 2009
"... This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal proce ..."
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Cited by 723 (18 self)
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This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo
sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rankone tensors, and the Tucker decomposition is a higherorder form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The Nway Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.
C.: Multichannel nonnegative matrix factorization in convolutive mixtures for audio source separation
 IEEE Trans. Audio, Speech, Language Process
, 2010
"... We consider inference in a general datadriven objectbased model of multichannel audio data, assumed generated as a possibly underdetermined convolutive mixture of source signals. Each source is given a model inspired from nonnegative matrix factorization (NMF) with the ItakuraSaito divergence, wh ..."
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Cited by 79 (17 self)
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We consider inference in a general datadriven objectbased model of multichannel audio data, assumed generated as a possibly underdetermined convolutive mixture of source signals. Each source is given a model inspired from nonnegative matrix factorization (NMF) with the ItakuraSaito divergence, which underlies a statistical model of superimposed Gaussian components. We address estimation of the mixing and source parameters using two methods. The first one consists of maximizing the exact joint likelihood of the multichannel data using an expectationmaximization algorithm. The second method consists of maximizing the sum of individual likelihoods of all channels using a multiplicative update algorithm inspired from NMF methodology. Our decomposition algorithms were applied to stereo music and assessed in terms of blind source separation performance. Index Terms — Multichannel audio, nonnegative matrix factorization, nonnegative tensor factorization, underdetermined convolutive blind source separation. 1.
ON TENSORS, SPARSITY, AND NONNEGATIVE FACTORIZATIONS
, 2012
"... 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 appropriat ..."
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Cited by 17 (1 self)
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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, both real and simulated.
Estimating the spatial position of spectral components in audio
 in Proc. Int. Conf. on Independent Component Analysis and Blind Source Separation (ICA
"... Abstract. One way of separating sources from a single mixture recording is by extracting spectral components and then combining them to form estimates of the sources. The grouping process remains a difficult problem. We propose, for instances when multiple mixture signals are available, clustering t ..."
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Abstract. One way of separating sources from a single mixture recording is by extracting spectral components and then combining them to form estimates of the sources. The grouping process remains a difficult problem. We propose, for instances when multiple mixture signals are available, clustering the components based on their relative contribution to each mixture (i.e., their spatial position). We introduce novel factorizations of magnitude spectrograms from multiple recordings and derive update rules that extend independent subspace analysis and nonnegative matrix factorization to concurrently estimate the spectral shape, time envelope and spatial position of each component. We show that estimated component positions are near the position of their corresponding source, and that multichannel nonnegative matrix factorization can distinguish three pianos by their position in the mixture. 1
Sparse nonnegative tensor factorization using columnwise coordinate descent
 Pattern Recognition
, 2012
"... Many applications in computer vision, biomedical informatics, and graphics deal with data in the matrix or tensor form. Nonnegative matrix and tensor factorization, which extract datadependent nonnegative basis functions, have been commonly applied for the analysis of such data for data compres ..."
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Cited by 13 (0 self)
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Many applications in computer vision, biomedical informatics, and graphics deal with data in the matrix or tensor form. Nonnegative matrix and tensor factorization, which extract datadependent nonnegative basis functions, have been commonly applied for the analysis of such data for data compression, visualization, and detection of hidden information (factors). In this paper, we present a fast and flexible algorithm for sparse nonnegative tensor factorization (SNTF) based on columnwise coordinate descent (CCD). Different from the traditional coordinate descent which updates one element at a time, CCD updates one column vector simultaneously. Our empirical results on highermode images, such as brain MRI images, gene expression images, and hyperspectral images show that the proposed algorithm is 12 orders of magnitude faster than several stateoftheart algorithms. Key words:
Notes on nonnegative tensor factorization of the spectrogram for audio source separation : statistical insights and towards selfclustering of the spatial cues
 in 7th International Symposium on Computer Music Modeling and Retrieval (CMMR
, 2010
"... Abstract. Nonnegative tensor factorization (NTF) of multichannel spectrograms under PARAFAC structure has recently been proposed by Fitzgerald et al as a mean of performing blind source separation (BSS) of multichannel audio data. In this paper we investigate the statistical source models implied by ..."
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Cited by 8 (4 self)
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Abstract. Nonnegative tensor factorization (NTF) of multichannel spectrograms under PARAFAC structure has recently been proposed by Fitzgerald et al as a mean of performing blind source separation (BSS) of multichannel audio data. In this paper we investigate the statistical source models implied by this approach. We show that it implicitly assumes a nonpointsource model contrasting with usual BSS assumptions and we clarify the links between the measure of fit chosen for the NTF and the implied statistical distribution of the sources. While the original approach of Fitzgeral et al requires a posterior clustering of the spatial cues to group the NTF components into sources, we discuss means of performing the clustering within the factorization. In the results section we test the impact of the simplifying nonpointsource assumption on underdetermined linear instantaneous mixtures of musical sources and discuss the limits of the approach for such mixtures.
Multichannel extensions of nonnegative matrix factorization with complexvalued data
 IEEE Transactions on Audio, Speech and Language Processing
, 2013
"... Abstract—This paper presents new formulations and algorithms for multichannel extensions of nonnegative matrix factorization (NMF). The formulations employ Hermitian positive semidefinite matrices to represent a multichannel version of nonnegative elements. Multichannel Euclidean distance and mult ..."
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Abstract—This paper presents new formulations and algorithms for multichannel extensions of nonnegative matrix factorization (NMF). The formulations employ Hermitian positive semidefinite matrices to represent a multichannel version of nonnegative elements. Multichannel Euclidean distance and multichannel ItakuraSaito (IS) divergence are defined based on appropriate statistical models utilizing multivariate complex Gaussian distributions. To minimize this distance/divergence, efficient optimization algorithms in the form of multiplicative updates are derived by using properly designed auxiliary functions. Two methods are proposed for clustering NMF bases according to the estimated spatial property. Convolutive blind source separation (BSS) is performed by the multichannel extensions of NMF with the clustering mechanism. Experimental results show that 1) the derived multiplicative update rules exhibited good convergence behavior, and 2) BSS tasks for several music sources with two microphones and three instrumental parts were evaluated successfully. Index Terms—Blind source separation, clustering, convolutive mixture, multichannel, nonnegative matrix factorization. I.
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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Cited by 7 (0 self)
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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
New formulations and efficient algorithms for multichannel NMF
 in Proc. WASPAA ’11
, 2011
"... This paper proposes new formulations and algorithms for a multichannel extension of nonnegative matrix factorization (NMF), intending convolutive sound source separation with multiple microphones. The proposed formulation employs Hermitian positive semidefinite matrices to represent a multichanne ..."
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Cited by 7 (3 self)
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This paper proposes new formulations and algorithms for a multichannel extension of nonnegative matrix factorization (NMF), intending convolutive sound source separation with multiple microphones. The proposed formulation employs Hermitian positive semidefinite matrices to represent a multichannel version of nonnegative elements. Such matrices are basically estimated for NMF bases, but a source separation task can be performed by introducing variables that relate NMF bases and sources. Efficient optimization algorithms in the form of multiplicative updates are derived by using properly designed auxiliary functions. Experimental results show that two instrumental sounds coming from different directions were successfully separated by the proposed algorithm. Index Terms — nonnegative matrix factorization, multichannel, positive semidefinite, auxiliary function, source separation 1.
Multichannel audio upmixing based on nonnegative tensor factorization representation
 In IEEE Workshop Applications of Signal Processing to Audio and Acoustics (WASPAA), New Paltz
, 2011
"... This paper proposes a new spatial audio coding (SAC) method that is based on parametrization of multichannel audio by sound objects using nonnegative tensor factorization (NTF). The spatial parameters are estimated using perceptually motivated NTF model and are used for upmixing a downmixed and en ..."
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Cited by 6 (0 self)
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This paper proposes a new spatial audio coding (SAC) method that is based on parametrization of multichannel audio by sound objects using nonnegative tensor factorization (NTF). The spatial parameters are estimated using perceptually motivated NTF model and are used for upmixing a downmixed and encoded mixture signal. The performance of the proposed coding is evaluated using listening tests, which prove the coding performance being on a par with conventional SAC methods. Additionally the proposed coding enables controlling the upmix content by meaningful objects. Index Terms — Spatial audio coding, Objectbased audio coding, Nonnegative tensor factorization