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24
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 228 (14 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.
Multiplying matrices faster than coppersmithwinograd
 In Proc. 44th ACM Symposium on Theory of Computation
, 2012
"... We develop new tools for analyzing matrix multiplication constructions similar to the CoppersmithWinograd construction, and obtain a new improved bound on ω < 2.3727. 1 ..."
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Cited by 39 (5 self)
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We develop new tools for analyzing matrix multiplication constructions similar to the CoppersmithWinograd construction, and obtain a new improved bound on ω < 2.3727. 1
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 15 (1 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Graph Expansion and Communication Costs of Fast Matrix Multiplication
"... The communication cost of algorithms (also known as I/Ocomplexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen’s and other fast matrix multiplication algorithms, and obtain the first lower bounds on their communi ..."
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Cited by 13 (11 self)
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The communication cost of algorithms (also known as I/Ocomplexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen’s and other fast matrix multiplication algorithms, and obtain the first lower bounds on their communication costs. For sequential algorithms these bounds are attainable and so optimal. 1.
Multivariate power series multiplication
 IN ISSAC’05
, 2005
"... We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M. In some special cases, such as partial degree truncation, this estima ..."
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Cited by 11 (6 self)
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We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M. In some special cases, such as partial degree truncation, this estimate carries over to total complexity. This leads to complexity improvements for some basic algorithms with algebraic numbers, and some polynomial system solving algorithms.
Breaking the CoppersmithWinograd barrier. Unpublished manuscript
, 2011
"... We develop new tools for analyzing matrix multiplication constructions similar to the CoppersmithWinograd construction, and obtain a new improved bound on ω < 2.3727. 1 ..."
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Cited by 11 (0 self)
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We develop new tools for analyzing matrix multiplication constructions similar to the CoppersmithWinograd construction, and obtain a new improved bound on ω < 2.3727. 1
The border rank of the multiplication of 2 × 2 matrices is seven
 J. Amer. Math. Soc
"... One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, ..."
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Cited by 10 (3 self)
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One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and
New lower bounds for the border rank of matrix multiplication
"... Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Li ..."
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Cited by 8 (2 self)
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Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3 2 n2 + n 2 − 1 for all n ≥ 3. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.
Example to “Determinantal equations for secant varieties and the EisenbudKohStillman conjecture
, 2012
"... Abstract. We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese reembeddings of arbitrary varieties. Eisenbud’s question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that settheoretic equations of small secant varietie ..."
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Cited by 6 (0 self)
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Abstract. We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese reembeddings of arbitrary varieties. Eisenbud’s question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that settheoretic equations of small secant varieties to a high degree Veronese reembedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counterexamples to the EKS conjecture for singular curves. The techniques we use also allow us to prove a gap and uniqueness theorem for symmetric tensor rank. We put Eisenbud’s question in a more general context about the behaviour of border rank under specialisation to a linear subspace, and provide an overview of conjectures coming from signal processing and complexity theory in this context. 1.