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14
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 TENSOR DECOMPOSITION
"... We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented ..."
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Cited by 6 (0 self)
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We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polynomial in n variables of total degree d. Thus the decomposition corresponds to a sum of powers of linear forms. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the tensor rank. 1.
Brambilla, Secant varieties of SegreVeronese varieties P m × P n embedded by O(1
 arXiv (2008), no. 0809.4837. ↑2 DANIEL ERMAN AND MAURICIO VELASCO
"... Abstract. Let Xm,n be the SegreVeronese variety P m × P n embedded by the morphism given by O(1,2). In this paper, we provide two functions s(m, n) ≤ s(m, n) such that the s th secant variety of Xm,n has the expected dimension if s ≤ s(m,n) or s(m, n) ≤ s. We also present a conjecturally complete ..."
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Cited by 5 (0 self)
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Abstract. Let Xm,n be the SegreVeronese variety P m × P n embedded by the morphism given by O(1,2). In this paper, we provide two functions s(m, n) ≤ s(m, n) such that the s th secant variety of Xm,n has the expected dimension if s ≤ s(m,n) or s(m, n) ≤ s. We also present a conjecturally complete list of defective secant varieties of such SegreVeronese varieties. 1.
Génération automatique de procédures numériques pour les fonctions Dfinies
"... L’évaluation numérique à grande précision de constantes comme π, e, γ, ln 2, etc., de fonctions élémentaires comme exp et arctan, puis de fonctions spéciales comme Γ ou ζ est un problème classique. D’un point de vue informatique, « grande précision » s’oppose à précision fixe, mais sousentend auss ..."
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Cited by 2 (1 self)
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L’évaluation numérique à grande précision de constantes comme π, e, γ, ln 2, etc., de fonctions élémentaires comme exp et arctan, puis de fonctions spéciales comme Γ ou ζ est un problème classique. D’un point de vue informatique, « grande précision » s’oppose à précision fixe, mais sousentend aussi que l’on cherche des algorithmes asymptotiquement efficaces quand le nombre de chiffres demandés grandit. Le développement d’algorithmes de complexité quasilinéaire en le nombre de chiffres du résultat remonte aux années 1970, avec par exemple les travaux de Richard Brent, Eugene Salamin ou R. William Gosper. Les fonctions holonomes sont les solutions d’équations différentielles linéaires à coefficients polynomiaux. Leurs propriétés élémentaires sont bien connues depuis le dixneuvième siècle, mais elles ont pris une place importante en combinatoire (comme séries génératrices) et en calcul formel (en tant que classe de fonctions bénéficiant de propriétés algorithmiques agréables, tant du point de vue de la calculabilité que de celui de la complexité) depuis les années 1980. Parmi les responsables de ce regain d’intérêt, on peut citer Richard Stanley, Leonard Lipshitz et Doron Zeilberger. Mon travail de stage s’incrit dans une démarche générale du projet Algo de développer pour toute la classe des fonctions holonomes une algorithmique efficace utilisant
The I/O complexity of sparse matrix dense matrix multiplication
 In Proceedings of LATIN’10
, 2010
"... We consider the multiplication of a sparse N × N matrix A with a dense N × N matrix B in the I/O model. We determine the worstcase nonuniform complexity of this task up to a constant factor for all meaningful choices of the parameters N (dimension of the matrices), k (average number of nonzero en ..."
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We consider the multiplication of a sparse N × N matrix A with a dense N × N matrix B in the I/O model. We determine the worstcase nonuniform complexity of this task up to a constant factor for all meaningful choices of the parameters N (dimension of the matrices), k (average number of nonzero entries per column or row in A, i.e., there are in total kN nonzero entries), M (main memory size), and B (block size), as long as M ≥ B 2 (tall cache assumption). For large and small k, the structure of the algorithm does not need to depend on the structure of the sparse matrix A, whereas for intermediate densities it is possible and necessary to find submatrices that fit in memory and are slightly denser than on average. The focus of this work is asymptotic worstcase complexity, i.e., the existence of matrices that require a certain number of I/Os and the existence of algorithms (sometimes depending on the shape of the sparse matrix) that use only a constant factor more I/Os. 1
RANKS OF TENSORS AND AND A GENERALIZATION OF SECANT VARIETIES
"... Abstract. We investigate differences between Xrank and Xborder rank, focusing on the cases of tensors and partially symmetric tensors. As an aid to our study, and as an object of interest in its own right, we define notions of Xrank and border rank for a linear subspace. Results include determini ..."
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Abstract. We investigate differences between Xrank and Xborder rank, focusing on the cases of tensors and partially symmetric tensors. As an aid to our study, and as an object of interest in its own right, we define notions of Xrank and border rank for a linear subspace. Results include determining and bounding the maximum Xrank of points in several cases of interest. 1.
GENERATION AND SYZYGIES OF THE FIRST SECANT VARIETY
, 809
"... Abstract. Under certain effective positivity conditions, we show that the secant variety to a smooth variety satisfies N3,p. For smooth curves, we provide the best possible effective bound on the degree d of the embedding, d ≥ 2g + 3 + p. 1. ..."
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Abstract. Under certain effective positivity conditions, we show that the secant variety to a smooth variety satisfies N3,p. For smooth curves, we provide the best possible effective bound on the degree d of the embedding, d ≥ 2g + 3 + p. 1.
Author manuscript, published in "Linear Algebra and Applications 433, 1112 (2010) 18511872" SYMMETRIC TENSOR DECOMPOSITION
, 2009
"... Abstract. We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variab ..."
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Abstract. We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring’s problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester’s approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasiHankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in nongeneric cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding
GENERAL TENSOR DECOMPOSITION, MOMENT MATRICES AND APPLICATIONS
, 2011
"... Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for fl ..."
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Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of QuasiHankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.