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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 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
, 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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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.
Secant varieties of SegreVeronese varieties P m × P n embedded by O(1,2)
, 2008
"... 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 ..."
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Cited by 18 (3 self)
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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.
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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Cited by 17 (3 self)
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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.
TENSOR COMPLEXES: MULTILINEAR FREE RESOLUTIONS CONSTRUCTED FROM HIGHER TENSORS
"... Abstract. The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1tensor), and the Eagon– Northcott and Buchsbaum–Rim complexes, which are constructed from a matrix (i.e., a 2tensor). The subject of this paper is a ..."
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Cited by 8 (5 self)
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Abstract. The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1tensor), and the Eagon– Northcott and Buchsbaum–Rim complexes, which are constructed from a matrix (i.e., a 2tensor). The subject of this paper is a multilinear analogue of these complexes, which we construct from an arbitrary higher tensor. Our construction provides detailed new examples of minimal free resolutions, as well as a unifying view on a wide variety of complexes including: the Eagon–Northcott, Buchsbaum– Rim and similar complexes, the Eisenbud–Schreyer pure resolutions, and the complexes used by Gelfand–Kapranov–Zelevinsky and Weyman to compute hyperdeterminants. In addition, we provide applications to the study of pure resolutions and Boij–Söderberg theory, including the construction of infinitely many new families of pure resolutions, and the first explicit description of the differentials of the Eisenbud–Schreyer pure resolutions. 1.
Multihomogeneous polynomial decomposition using moment matrices
 International Symposium on Symbolic and Algebraic Computation
, 2011
"... 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 7 (4 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.
A note on compressed sensing and the complexity of matrix multiplication
 Inf. Process. Lett 109
, 2009
"... We consider the conjectured O(N2+) time complexity of multiplying any two N × N matrices A and B. Our main result is a deterministic Compressed Sensing (CS) algorithm that both rapidly and accurately computes A · B provided that the resulting matrix product is sparse/compressible. As a consequence ..."
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Cited by 5 (0 self)
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We consider the conjectured O(N2+) time complexity of multiplying any two N × N matrices A and B. Our main result is a deterministic Compressed Sensing (CS) algorithm that both rapidly and accurately computes A · B provided that the resulting matrix product is sparse/compressible. As a consequence of our main result we increase the class of matrices A, for any given N × N matrix B, which allows the exact computation of A · B to be carried out using the conjectured O(N2+) operations. Additionally, in the process of developing our matrix multiplication procedure, we present a modified version of Indyk’s recently proposed extractorbased CS algorithm [12] which is resilient to noise. Key words: algorithms, analysis of algorithms, approximation algorithms, computational complexity 1