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Tensor Decompositions and Applications

by Tamara G. Kolda, Brett W. Bader - SIAM REVIEW , 2009
"... This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N -way array. Decompositions of higher-order tensors (i.e., N -way arrays with N ≥ 3) have applications in psychometrics, chemometrics, signal proce ..."
Abstract - Cited by 723 (18 self) - Add to MetaCart
processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decompo- sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum

Pfister Involutions

by E Bayer-Fluckiger, R Parimala, A Quéguiner-mathieu , 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 ..."
Abstract - Cited by 312 (21 self) - Add to MetaCart
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, High-Performance, ANSI C Coding Methodology

by Jeff Bilmes, Krste Asanovic , Chee-Whye Chin , Jim Demmel , 1996
"... Modern microprocessors can achieve high performance on linear algebra kernels but this currently requires extensive machine-specific hand tuning. We have developed a methodology whereby near-peak performance on a wide range of systems can be achieved automatically for such routines. First, by analyz ..."
Abstract - Cited by 268 (24 self) - Add to MetaCart
Modern microprocessors can achieve high performance on linear algebra kernels but this currently requires extensive machine-specific hand tuning. We have developed a methodology whereby near-peak performance on a wide range of systems can be achieved automatically for such routines. First

Multilinear Analysis of Image Ensembles: TensorFaces

by M. Alex O. Vasilescu, Demetri Terzopoulos - 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 higher-order tensors, offers a potent mathematical framework for analyzing the multifactor structure of image ensembles and for addressing the d ..."
Abstract - Cited by 188 (7 self) - Add to MetaCart
Natural images are the composite consequence of multiple factors related to scene structure, illumination, and imaging. Multilinear algebra, the algebra of higher-order tensors, offers a potent mathematical framework for analyzing the multifactor structure of image ensembles and for addressing

TENSOR RANK AND THE ILL-POSEDNESS OF THE BEST LOW-RANK APPROXIMATION PROBLEM

by Vin De Silva, Lek-heng Lim
"... There has been continued interest in seeking a theorem describing optimal low-rank 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 ..."
Abstract - Cited by 194 (13 self) - Add to MetaCart
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)

by J. A. de Azcárraga, A. J. Macfarlane , 2000
"... ..."
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Orthogonal Tensor Decompositions

by Tamara G. Kolda - SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS , 2001
"... We explore the orthogonal decomposition of tensors (also known as multidimensional arrays or n-way 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 ..."
Abstract - Cited by 124 (9 self) - Add to MetaCart
tensor extension of the Eckart-Young SVD approximation theorem by Leibovici and Sabatier [Linear Algebra Appl., 269 (1998), pp. 307-329].

Symmetric tensors and symmetric tensor rank

by Pierre Comon, Gene Golub, Lek-heng Lim, Bernard Mourrain - 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 rank-1 order-k tensor is the outer product of k non-zero vectors. An ..."
Abstract - Cited by 99 (20 self) - Add to MetaCart
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 rank-1 order-k tensor is the outer product of k non-zero vectors

On the Morita equivalence of tensor algebras

by Paul S. Muhly, Baruch Solel - Proc. London Math. Soc
"... Our objective is two fold. First, we want to develop a notion of Morita equivalence for C-correspondences that guarantees that if two C-correspondences 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 ..."
Abstract - Cited by 42 (16 self) - Add to MetaCart
Our objective is two fold. First, we want to develop a notion of Morita equivalence for C-correspondences that guarantees that if two C-correspondences 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

by Yi-zhi Huang, James Lepowsky - 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 ..."
Abstract - Cited by 68 (13 self) - Add to MetaCart
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 of 3,962