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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 237 (14 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.
Blind identification of underdetermined mixtures based on the characteristic function
 Signal Process
, 2005
"... Linear mixtures of independent random variables (the socalled sources) are sometimes referred to as underdetermined mixtures (UDM) when the number of sources exceeds the dimension of the observation space. The algorithms proposed are able to identify algebraically a UDM using the second characteri ..."
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Cited by 28 (16 self)
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Linear mixtures of independent random variables (the socalled sources) are sometimes referred to as underdetermined mixtures (UDM) when the number of sources exceeds the dimension of the observation space. The algorithms proposed are able to identify algebraically a UDM using the second characteristic function (c.f.) of the observations, without any need of sparsity assumption on sources. In fact, by taking higher order derivatives of the multivariate c.f. core equation, the blind identification problem is shown to reduce to a tensor decomposition. With only two sensors, the first algorithm only needs a SVD. With a larger number of sensors, the second algorithm executes an alternating least squares (ALS) algorithm. The joint use of statistics of different orders is possible, and a LS solution can be computed. Identifiability conditions are stated in each of the two cases. Computer simulations eventually demonstrate performances in the absence of sparsity, and emphasize the interest in using jointly derivatives of different orders. r 2005 Elsevier B.V. All rights reserved.
Blind identification of underdetermined mixtures by simultaneous matrix diagonalization
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2008
"... In this paper, we study simultaneous matrix diagonalizationbased techniques for the estimation of the mixing matrix in underdetermined independent component analysis (ICA). This includes a generalization to underdetermined mixtures of the wellknown SOBI algorithm. The problem is reformulated in t ..."
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Cited by 12 (2 self)
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In this paper, we study simultaneous matrix diagonalizationbased techniques for the estimation of the mixing matrix in underdetermined independent component analysis (ICA). This includes a generalization to underdetermined mixtures of the wellknown SOBI algorithm. The problem is reformulated in terms of the parallel factor decomposition (PARAFAC) of a higherorder tensor. We present conditions under which the mixing matrix is unique and discuss several algorithms for its computation.
Tensor Algebra and Multidimensional Harmonic Retrieval in Signal Processing for MIMO Radar
"... Abstract—Detection and estimation problems in multipleinput multipleoutput (MIMO) radar have recently drawn considerable interest in the signal processing community. Radar has long been a staple of signal processing, and MIMO radar presents challenges and opportunities in adapting classical radar ..."
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Cited by 11 (1 self)
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Abstract—Detection and estimation problems in multipleinput multipleoutput (MIMO) radar have recently drawn considerable interest in the signal processing community. Radar has long been a staple of signal processing, and MIMO radar presents challenges and opportunities in adapting classical radar imaging tools and developing new ones. Our aim in this article is to showcase the potential of tensor algebra and multidimensional harmonic retrieval (HR) in signal processing for MIMO radar. Tensor algebra and multidimensional HR are relatively mature topics, albeit still on the fringes of signal processing research. We show they are in fact central for target localization in a variety of pertinent MIMO radar scenarios. Tensor algebra naturally comes into play when the coherent processing interval comprises multiple pulses, or multiple transmit and receive subarrays are used (multistatic configuration). Multidimensional harmonic structure emerges for farfield uniform linear transmit/receive array configurations, also taking into account Doppler shift; and hybrid models arise inbetween. This viewpoint opens the door for the application and further development of powerful algorithms and identifiability results for MIMO radar. Compared to the classical radarimagingbased methods such as Capon or MUSIC, these algebraic techniques yield improved performance, especially for closely spaced targets, at modest complexity. Index Terms—DoADoD estimation, harmonic retrieval, localization, multipleinput multipleoutput (MIMO) radar, tensor decomposition. I.
Tensor Decompositions, Alternating Least Squares and Other Tales
 JOURNAL OF CHEMOMETRICS
, 2009
"... This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple “bottlenecks”, and on “swamps”. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity ..."
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Cited by 11 (3 self)
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This work was originally motivated by a classification of tensors proposed by Richard Harshman. In particular, we focus on simple and multiple “bottlenecks”, and on “swamps”. Existing theoretical results are surveyed, some numerical algorithms are described in details, and their numerical complexity is calculated. In particular, the interest in using the ELS enhancement in these algorithms is discussed. Computer simulations feed this discussion.
Adaptive Algorithms to Track the PARAFAC Decomposition of a ThirdOrder Tensor
"... Abstract—The PARAFAC decomposition of a higherorder tensor is a powerful multilinear algebra tool that becomes more and more popular in a number of disciplines. Existing PARAFAC algorithms are computationally demanding and operate in batch mode—both serious drawbacks for online applications. When ..."
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Cited by 3 (2 self)
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Abstract—The PARAFAC decomposition of a higherorder tensor is a powerful multilinear algebra tool that becomes more and more popular in a number of disciplines. Existing PARAFAC algorithms are computationally demanding and operate in batch mode—both serious drawbacks for online applications. When the data are serially acquired, or the underlying model changes with time, adaptive PARAFAC algorithms that can track the sought decomposition at low complexity would be highly desirable. This is a challenging task that has not been addressed in the literature, and the topic of this paper. Given an estimate of the PARAFAC decomposition of a tensor at instant t, we propose two adaptive algorithms to update the decomposition at instant t +1, the new tensor being obtained from the old one after appending a new slice in the ’time ’ dimension. The proposed algorithms can yield estimation performance that is very close to that obtained via repeated application of stateofart batch algorithms, at orders of magnitude lower complexity. The effectiveness of the proposed algorithms is illustrated using a MIMO radar application (tracking of directions of arrival and directions of departure) as an example. Index Terms—Adaptive algorithms, DOA/DOD tracking, higherorder tensor, MIMO radar, PARAllel FACtor (PARAFAC).
Lathauwer, “Line search computation of the block factor model for blind multiuser access in wireless communications,” presented at the
 IEEE Workshop Signal Processing Advances Wireless Communications (SPAWC
"... In this paper, we present a technique for the blind separation of DSCDMA signals received on an antenna array, for a multipath propagation scenario that generates InterSymbolInterference. Our method relies on a new thirdorder tensor decomposition, which is a generalization of the parallel facto ..."
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Cited by 3 (1 self)
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In this paper, we present a technique for the blind separation of DSCDMA signals received on an antenna array, for a multipath propagation scenario that generates InterSymbolInterference. Our method relies on a new thirdorder tensor decomposition, which is a generalization of the parallel factor model. We start from the observation that the temporal, spatial and spectral diversities give a thirdorder tensor structure to the received data. This tensor is then decomposed in a sum of contributions, where each contribution fully characterizes one user. We also present a Line Search scheme that greatly improves the convergence speed of the alternating least squares algorithm previously used. 1.
A nonlinear GMRES optimization algorithm for canonical tensor decomposition
 SIAM Journal on Scientific Computing accepted
, 2012
"... Abstract. A new algorithm is presented for computing a canonical rankR tensor approximation that has minimal distance to a given tensor in the Frobenius norm, where the canonical rankR tensor consists of the sum of R rankone tensors. Each iteration of the method consists of three steps. In the fi ..."
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Cited by 2 (2 self)
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Abstract. A new algorithm is presented for computing a canonical rankR tensor approximation that has minimal distance to a given tensor in the Frobenius norm, where the canonical rankR tensor consists of the sum of R rankone tensors. Each iteration of the method consists of three steps. In the first step, a tentative new iterate is generated by a standalone onestep process, for which we use alternating least squares (ALS). In the second step, an accelerated iterate is generated by a nonlinear generalized minimal residual (GMRES) approach, recombining previous iterates in an optimal way, and essentially using the standalone onestep process as a preconditioner. In particular, the nonlinear extension of GMRES we use that was proposed by Washio and Oosterlee in [Electron. Trans. Numer. Anal., 15 (2003), pp. 165–185] for nonlinear partial differential equation problems (which is itself related to other existing acceleration methods for nonlinear equation systems). In the third step, a line search is performed for globalization. The resulting nonlinear GMRES (NGMRES) optimization algorithm is applied to dense and sparse tensor decomposition test problems. The numerical tests show that ALS accelerated by NGMRES may significantly outperform standalone ALS when highly accurate stationary points are desired for difficult problems. Further comparison tests show that NGMRES is competitive with the wellknown nonlinear conjugate gradient method for the test problems considered and outperforms it in many cases. The proposed NGMRES optimization algorithm is based on general concepts and may be applied to other nonlinear optimization problems. Key words. optimization canonical tensor decomposition, alternating least squares, GMRES, nonlinear
Batch and Adaptive PARAFACBased Blind Separation of Convolutive Speech Mixtures
"... Abstract—We present a frequencydomain technique based on PARAllel FACtor (PARAFAC) analysis that performs multichannel blind source separation (BSS) of convolutive speech mixtures. PARAFAC algorithms are combined with a dimensionality reduction step to significantly reduce computational complexity. ..."
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Cited by 2 (1 self)
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Abstract—We present a frequencydomain technique based on PARAllel FACtor (PARAFAC) analysis that performs multichannel blind source separation (BSS) of convolutive speech mixtures. PARAFAC algorithms are combined with a dimensionality reduction step to significantly reduce computational complexity. The identifiability potential of PARAFAC is exploited to derive a BSS algorithm for the underdetermined case (more speakers than microphones), combining PARAFAC analysis with timevarying Capon beamforming. Finally, a lowcomplexity adaptive version of the BSS algorithm is proposed that can track changes in the mixing environment. Extensive experiments with realistic and measured data corroborate our claims, including the underdetermined case. Signaltointerference ratio improvements of up to 6 dB are shown compared to stateoftheart BSS algorithms, at an order of magnitude lower computational complexity. Index Terms—Adaptive separation, blind speech separation,, joint diagonalization, PARAllel FACtor (PARAFAC), permutation ambiguity, underdetermined case.
NONITERATIVE SOLUTION FOR PARAFAC WITH A TOEPLITZ MATRIX FACTOR
"... Recently, tensor signal processing has received an increased attention, particularly in the context of wireless communication applications. The socalled PARAllel FACtor (PARAFAC) decomposition is certainly the most used tensor tool. In general, the parameter estimation of a PARAFAC decomposition is ..."
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Cited by 2 (0 self)
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Recently, tensor signal processing has received an increased attention, particularly in the context of wireless communication applications. The socalled PARAllel FACtor (PARAFAC) decomposition is certainly the most used tensor tool. In general, the parameter estimation of a PARAFAC decomposition is carried out by means of the iterative ALS algorithm, which exhibits the following main drawbacks: convergence towards local minima, a high number of iterations for convergence, and difficulty to take, optimally, special matrix structures into account. In this paper, we propose a noniterative parameter estimation method for a PARAFAC decomposition when one matrix factor has a Toeplitz structure, a situation that is commonly encountered in signal processing applications. We illustrate the proposed method by means of simulation results. 1.