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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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Cited by 723 (18 self)
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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
Pfister Involutions
, 2003
"... The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that ..."
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Cited by 312 (21 self)
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The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven
Optimizing Matrix Multiply using PHiPAC: a Portable, HighPerformance, ANSI C Coding Methodology
, 1996
"... Modern microprocessors can achieve high performance on linear algebra kernels but this currently requires extensive machinespecific hand tuning. We have developed a methodology whereby nearpeak performance on a wide range of systems can be achieved automatically for such routines. First, by analyz ..."
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Cited by 268 (24 self)
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Modern microprocessors can achieve high performance on linear algebra kernels but this currently requires extensive machinespecific hand tuning. We have developed a methodology whereby nearpeak performance on a wide range of systems can be achieved automatically for such routines. First
Multilinear Analysis of Image Ensembles: TensorFaces
 IN PROCEEDINGS OF THE EUROPEAN CONFERENCE ON COMPUTER VISION
, 2002
"... Natural images are the composite consequence of multiple factors related to scene structure, illumination, and imaging. Multilinear algebra, the algebra of higherorder tensors, offers a potent mathematical framework for analyzing the multifactor structure of image ensembles and for addressing the d ..."
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Cited by 188 (7 self)
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Natural images are the composite consequence of multiple factors related to scene structure, illumination, and imaging. Multilinear algebra, the algebra of higherorder tensors, offers a potent mathematical framework for analyzing the multifactor structure of image ensembles and for addressing
TENSOR RANK AND THE ILLPOSEDNESS OF THE BEST LOWRANK APPROXIMATION PROBLEM
"... There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, te ..."
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Cited by 194 (13 self)
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for extending results upwards to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; in other words, can be determined by a system of polynomial inequalities. We
Compilation of relations for the antisymmetric tensors defined by the Lie algebra cocycles of su(n)
, 2000
"... ..."
Orthogonal Tensor Decompositions
 SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
, 2001
"... We explore the orthogonal decomposition of tensors (also known as multidimensional arrays or nway arrays) using two different definitions of orthogonality. We present numerous examples to illustrate the difficulties in understanding such decompositions. We conclude with a counterexample to a tensor ..."
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Cited by 124 (9 self)
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tensor extension of the EckartYoung SVD approximation theorem by Leibovici and Sabatier [Linear Algebra Appl., 269 (1998), pp. 307329].
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 99 (20 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
On the Morita equivalence of tensor algebras
 Proc. London Math. Soc
"... Our objective is two fold. First, we want to develop a notion of Morita equivalence for Ccorrespondences that guarantees that if two Ccorrespondences E and F are Morita equivalent, then the tensor algebras of E and F, T
E and T
F , are strongly Morita equivalent in the sense of [8], the Toepl ..."
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Cited by 42 (16 self)
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Our objective is two fold. First, we want to develop a notion of Morita equivalence for Ccorrespondences that guarantees that if two Ccorrespondences E and F are Morita equivalent, then the tensor algebras of E and F, T
E and T
F , are strongly Morita equivalent in the sense of [8
Tensor products of modules for a vertex operator algebras and vertex tensor categories
 in: Lie Theory and Geometry, in honor of Bertram Kostant
, 1994
"... In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announ ..."
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Cited by 68 (13 self)
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In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5
Results 1  10
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