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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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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.
Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
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 33 (9 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.
Generic and typical ranks of multiway arrays
 Linear Algebra Appl
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 27 (5 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Decomposing tensors with structured matrix factors reduces to rank1 approximations
 in IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP
, 2010
"... tensors with structured matrix factors reduces to rank1 approximations ..."
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Cited by 9 (1 self)
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tensors with structured matrix factors reduces to rank1 approximations
Generic and typical ranks of threeway arrays
 Research Report ISRN I3S/RR200629FR, I3S, SophiaAntipolis
"... The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out Factor Analysis on arrays with more than two indices. The generic rank may be seen as an upper bound to the number of factors that can be extracted from a giv ..."
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Cited by 6 (3 self)
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The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out Factor Analysis on arrays with more than two indices. The generic rank may be seen as an upper bound to the number of factors that can be extracted from a given tensor. We explain in this short paper how to obtain numerically the generic rank of tensors of arbitrary dimensions, and compare it with the rare algebraic results already known at order three. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, or tensors with free entries. Résumé La notion de rang tensoriel, proposée dans les années vingt, a été popularisée au début des années soixantedix. Ceci a permis de mettre en oeuvre l’Analyse de Facteurs sur des tableaux de données comportant plus de deux indices. Le rang générique peut être vu comme une borne supérieure sur le nombre de facteurs pouvant être extraits d’un tenseur donné. Nous expliquons dans ce court article comment trouver numériquement le rang générique d’un tenseur de dimensions arbitraires, et le comparons aux quelques rares résultats algébriques déjà connus à l’ordre trois. Nous examinons notamment les cas des tenseurs symétriques, des tenseurs à tranches matricielles symétriques, ou des tenseurs à éléments libres.
On the typical rank of real binary forms
, 2009
"... We determine the rank of a general real binary form of degree d = 4 and d = 5. In the case d = 5, the possible values of the rank of such general forms are 3, 4, 5. The existence of three typical ranks was unexpected. We prove that a real binary form of degree d with d real roots has rank d. ..."
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Cited by 6 (1 self)
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We determine the rank of a general real binary form of degree d = 4 and d = 5. In the case d = 5, the possible values of the rank of such general forms are 3, 4, 5. The existence of three typical ranks was unexpected. We prove that a real binary form of degree d with d real roots has rank d.
Decoupling multivariate polynomials using firstorder information and tensor decompositions
 SIAM J. Matrix Anal. Appl
, 2015
"... Abstract. We present a method to decompose a set of multivariate real polynomials into linear combinations of univariate polynomials in linear forms of the input variables. The method proceeds by collecting the firstorder information of the polynomials in a set of sampling points, which is captured ..."
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Abstract. We present a method to decompose a set of multivariate real polynomials into linear combinations of univariate polynomials in linear forms of the input variables. The method proceeds by collecting the firstorder information of the polynomials in a set of sampling points, which is captured by the Jacobian matrix evaluated at the sampling points. The canonical polyadic decomposition of the threeway tensor of Jacobian matrices directly returns the unknown linear relations as well as the necessary information to reconstruct the univariate polynomials. The conditions under which this decoupling procedure works are discussed, and the method is illustrated on several numerical examples.
Some Numerical Results on the Rank of Generic ThreeWay Arrays over R
, 2009
"... Abstract. The aim of this paper is the introduction of a new method for the numerical computation of the rank of a threeway array, X ∈ RI×J×K, over R. We show that the rank of a threeway array over R is intimately related to the real solution set of a system of polynomial equations. Using this, we ..."
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Abstract. The aim of this paper is the introduction of a new method for the numerical computation of the rank of a threeway array, X ∈ RI×J×K, over R. We show that the rank of a threeway array over R is intimately related to the real solution set of a system of polynomial equations. Using this, we present some numerical results based on the computation of Gröbner bases.