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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 705 (17 self)
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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
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 122 (9 self)
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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
Are Tensor Decomposition Solutions Unique?
, 902
"... For tensor decompositions such as HOSVD and ParaFac, the objective functions are nonconvex. This implies, theoretically, there exists a large number of local optimas: starting from different starting point, the iteratively improved solution will converge to different local solutions. This nonunique ..."
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For tensor decompositions such as HOSVD and ParaFac, the objective functions are nonconvex. This implies, theoretically, there exists a large number of local optimas: starting from different starting point, the iteratively improved solution will converge to different local solutions. This non
Smoothed Analysis of Tensor Decompositions
"... Low rank decomposition of tensors is a powerful tool for learning generative models. The uniqueness of decomposition gives tensors a significant advantage over matrices. However, tensors pose significant algorithmic challenges and tensors analogs of much of the matrix algebra toolkit are unlikely t ..."
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Cited by 3 (1 self)
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Low rank decomposition of tensors is a powerful tool for learning generative models. The uniqueness of decomposition gives tensors a significant advantage over matrices. However, tensors pose significant algorithmic challenges and tensors analogs of much of the matrix algebra toolkit are unlikely
Tensor Decomposition for Fast Parsing with
"... We describe an approach to speedup inference with latentvariable PCFGs, which have been shown to be highly effective for natural language parsing. Our approach is based on a tensor formulation recently introduced for spectral estimation of latentvariable PCFGs coupled with a tensor decomposition ..."
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We describe an approach to speedup inference with latentvariable PCFGs, which have been shown to be highly effective for natural language parsing. Our approach is based on a tensor formulation recently introduced for spectral estimation of latentvariable PCFGs coupled with a tensor decomposition
Canonical Tensor Decompositions
 ARCC WORKSHOP ON TENSOR DECOMPOSITION
, 2004
"... The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short pap ..."
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Cited by 42 (16 self)
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The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the HighOrder SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short
SYMMETRIC TENSOR DECOMPOSITION
, 2009
"... 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 30 (5 self)
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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
Statistical Performance of Convex Tensor Decomposition
"... We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as nonconvex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared erro ..."
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Cited by 35 (5 self)
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We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as nonconvex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared
Giorgio Ottaviani Tutorial on Tensor rank and tensor decomposition
"... Tutorial: A brief survey on tensor rank and tensor decomposition, from a geometric perspective. Workshop ..."
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Tutorial: A brief survey on tensor rank and tensor decomposition, from a geometric perspective. Workshop
Distributed Computation of Tensor Decompositions
 in Collaborative Networks,” Proc. IEEE CAMSAP 2013, SaintMartin
, 2013
"... 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 3 (2 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.
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