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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 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.
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.
Optimization Techniques for Small Matrix Multiplication
, 2010
"... The complexity of matrix multiplication has attracted a lot of attention in the last forty years. In this paper, instead of considering asymptotic aspects of this problem, we are interested in reducing the cost of multiplication for matrices of small size, say up to 30. Following previous work in a ..."
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Cited by 3 (0 self)
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The complexity of matrix multiplication has attracted a lot of attention in the last forty years. In this paper, instead of considering asymptotic aspects of this problem, we are interested in reducing the cost of multiplication for matrices of small size, say up to 30. Following previous work in a similar vein by Probert & Fischer, Smith, and Mezzarobba, we base our approach on previous algorithms for small matrices, due to Strassen, Winograd, Pan, Laderman,... and show how to exploit these standard algorithms in an improved way. We illustrate the use of our results by generating multiplication code over various rings, such as integers, polynomials, differential operators or linear recurrence operators. Keywords: matrix multiplication, small matrix, complexity.
A NOTE ON THE MULTIPLICATION OF TWO 3 X 3 FIBONACCIROWED MATRICES
"... A Fibonaccirowed matrix is defined to be a matrix in which each row consists of consecutive Fibonacci numbers in increasing order. Laderman [1] presented a noncommutative algorithm for multiplying two 3 x 3 matrices using 23 multiplications. It still needs 18 multiplications if Laderman 1 s algorit ..."
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A Fibonaccirowed matrix is defined to be a matrix in which each row consists of consecutive Fibonacci numbers in increasing order. Laderman [1] presented a noncommutative algorithm for multiplying two 3 x 3 matrices using 23 multiplications. It still needs 18 multiplications if Laderman 1 s algorithm is applied to the product of two 3 x 3 Fibonaccirowed matrices. In this short note, an algorithm is developed in which only 17 multiplications are needed. This algorithm is mainly based on Strassen f s result [2] and the fact that the third column of a Fibonaccirowed matrix is equal to the sum of the other two columns. Let C = AB be the matrix of the multiplication of two 3 x 3 Fibonaccirowed matrices. Define I = (alx + a22) (fclx + b22) II = a23b11 III = a11(b12 b22) IV = a22(Z?11 + b21) V = a13b22
Complexity of Bilinear Problems
"... 1 Computations and costs 7 1.1 Karatsuba’s algorithm............................... 7 1.2 A general model.................................. 8 ..."
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1 Computations and costs 7 1.1 Karatsuba’s algorithm............................... 7 1.2 A general model.................................. 8