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A Lower Bound for Matrix Multiplication
 SIAM J. Comput
, 1988
"... We prove that computing the product of two n × n matrices over the binary field requires at least 2.5 n 2  o ( n 2 ) multiplications. Key Words : matrix multiplication, arithmetic complexity, lower bounds, linear codes. 1. INTRODUCTION Let x = ( x 1 , . . . , x n ) T and y = ( y 1 , . . . , y ..."
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Cited by 17 (2 self)
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We prove that computing the product of two n × n matrices over the binary field requires at least 2.5 n 2  o ( n 2 ) multiplications. Key Words : matrix multiplication, arithmetic complexity, lower bounds, linear codes. 1. INTRODUCTION Let x = ( x 1 , . . . , x n ) T and y = ( y 1 , . . . , y m ) T be column vectors of indeterminates. A straightline algorithm for computing a set of bilinear forms in x and y is called quadratic ( respectively bilinear ), if all its nonscalar multiplication are of the shape l ( x , y ) . l ( x , y ) , (respectively l ( x ) . l ( y ) ) where l and l are linear forms of the indeterminates. 1 In this paper we establish the new 2.5 n 2  o ( n 2 ) lower bound on the multiplicative complexity of quadratic algorithms for multiplying n × n matrices over the binary field Z 2 . Let M F ( n , m , k ) and M ## F ( n , m , k ) denote the number of multiplications required to compute the product of n ×m and m ×k matrices by means of quadratic ...
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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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.
Some Computational Problems in Linear Algebra as Hard as Matrix Multiplication
"... We define the complexity of a computational problem given by a relation using the model of a computation tree with Ostrowski complexity measure. To a sequence of problems we assign an exponent similar as for matrix multiplication. For the complexity of the following computational problems in linear ..."
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We define the complexity of a computational problem given by a relation using the model of a computation tree with Ostrowski complexity measure. To a sequence of problems we assign an exponent similar as for matrix multiplication. For the complexity of the following computational problems in linear algebra ffl KER n : Compute a basis of the kernel for a given n \Theta nmatrix. ffl OGB n : Find an invertible matrix that transforms a given symmetric n \Theta n matrix to diagonal form. ffl SPR n : Find a sparse representation of a given n \Theta nmatrix. we prove relative lower bounds of the form aM n \Gamma b and absolute lower bounds dn 2 , where M n denotes the complexity of matrix multiplication and a; b; d are suitably chosen constants. We show that the exponent of the problem sequences KER; OGB; SPR is the same as the exponent ! of matrix multiplication. Supported in part by the Leibniz Center for Research in Computer Science, by the DFG Grant KA 673/21 and by the ...