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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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Cited by 101 (22 self)
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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.
Genericity and Rank Deficiency of High Order Symmetric Tensors
 Proc. IEEE Int. Conference on Acoustics, Speech, and Signal Processing (ICASSP
, 2006
"... Blind Identification of UnderDetermined Mixtures (UDM) is involved in numerous applications, including MultiWay factor Analysis (MWA) and Signal Processing. In the latter case, the use of HighOrder Statistics (HOS) like Cumulants leads to the decomposition of symmetric tensors. Yet, little has be ..."
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Cited by 9 (6 self)
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Blind Identification of UnderDetermined Mixtures (UDM) is involved in numerous applications, including MultiWay factor Analysis (MWA) and Signal Processing. In the latter case, the use of HighOrder Statistics (HOS) like Cumulants leads to the decomposition of symmetric tensors. Yet, little has been published about rankrevealing decompositions of symmetric tensors. Definitions of rank are discussed, and useful results on Generic Rank are proved, with the help of tools borrowed from Algebraic Geometry. 1.
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.
TENSOR MODELING AND SIGNAL PROCESSING FOR WIRELESS COMMUNICATION SYSTEMS
, 2010
"... Sciences et Technologies de l’Information et de la Communication ..."
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Cited by 1 (0 self)
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Sciences et Technologies de l’Information et de la Communication
Flow cytometry is an investigati...
, 2014
"... The paper presents a novel approach to the processing of flow cytometry data sequences. It consists in decomposing a sequence of multidimensional probability density functions by using multilinear block tensor decomposition approach [1], [2]. Also a formal link between flow cytometry data and fluore ..."
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The paper presents a novel approach to the processing of flow cytometry data sequences. It consists in decomposing a sequence of multidimensional probability density functions by using multilinear block tensor decomposition approach [1], [2]. Also a formal link between flow cytometry data and fluorescence spectra is provided allowing the joint processing of both data. To illustrate the effectiveness of the approach, a study of the T47D cell line mitochondrial membrane potential as a function of the CCCP decoupling agent concentration is performed. The main advantages of the method are: (i) the flow cytometry data compensation is no longer necessary, (ii) the cell sorting capacity of the method is significantly improved as compared to classical clustering methods. As a byproduct, it was possible to observe directly on the result of the processing, the dependence of the cell mitochondrial membrane potential with respect to the cell cycle phase. The proposed method is quite general provided that it is possible to design an experiment allowing to observe the response of cell populations to an environmental/chemical/biological parameter. Index Terms Flow cytometry, fluorescence spectroscopy, mixture of multivariate probability density functions, nonnegative block Candecomp/Parafac decomposition, nonnegative matrix factorization, mitochondrial membrane potential, JC1 probe. ∗ These authors contributed equally to this work.
The Constrained Trilinear Decomposition With Application to MIMO Wireless Communication Systems
"... du troisième ordre en une somme triple de facteurs tensoriels de rang1 avec des intéractions entre les différents facteurs. La structure d’intéraction est contrôlée par trois matrices de contraintes composées par des vecteurs canoniques. Nous présentons une application de cette décompositi ..."
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du troisième ordre en une somme triple de facteurs tensoriels de rang1 avec des intéractions entre les différents facteurs. La structure d’intéraction est contrôlée par trois matrices de contraintes composées par des vecteurs canoniques. Nous présentons une application de cette décomposition a ̀ des systèmes de communication sansfil MIMO (MultipleInput MultipleOutput). Une nouvelle structure de transmission est proposée, ou ̀ les matrices de contraintes de la décomposition sont exploitées pour configurer des précodeurs canoniques. La détection aveugle est possible grâce aux propriétés d’unicite ́ partielle de la décomposition. Pour illustrer cette application, le taux d’erreur de bit est évalue ́ pour quelques choix de précodeurs. Abstract – In this paper, we present a new tensor decomposition that consists in decomposing a thirdorder tensor into a triple sum of rankone tensor factors, where interactions involving the components of different factors are allowed. The interaction pattern is controlled by three constraint matrices composed of canonical vectors. An application of this decomposition to MultipleInput MultipleOutput (MIMO) wireless communication systems is presented. A new multipleantenna transmission structure is proposed, where the constraint matrices of the decomposition are exploited to design canonical precoders. Blind detection is possible thanks to the partial uniqueness properties of the decomposition. For illustrating this application, we evaluate the biterrorrate performance for some precoder configurations. 1
présentée et soutenue par
, 2010
"... Tensor modeling and signal processing for wireless communication systems Andre ́ De Almeida To cite this version: Andre ́ De Almeida. Tensor modeling and signal processing for wireless communication systems. ..."
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Tensor modeling and signal processing for wireless communication systems Andre ́ De Almeida To cite this version: Andre ́ De Almeida. Tensor modeling and signal processing for wireless communication systems.
Generic and Typical Ranks of . . .
, 2009
"... The concept of tensor rank was introduced in the twenties. In the seventies, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to con ..."
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The concept of tensor rank was introduced in the twenties. In the seventies, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor. We explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions, based on Terracini’s lemma, and compare it with the algebraic results already known in the real or complex fields. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, complex tensors enjoying Hermitian symmetries, or merely tensors with free entries.