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ON THE IDEALS OF SECANT VARIETIES OF SEGRE VARIETIES
, 2003
"... We establish basic techniques for studying the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture settheoretically for an arbitrary n ..."
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Cited by 40 (8 self)
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We establish basic techniques for studying the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture settheoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.
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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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.
Strassen’s Matrix Multiplication on GPUs
"... Abstract—We provide efficient singleprecision and integer GPU implementations of Strassen’s algorithm as well as of Winograd’s variant. On an NVIDIA C1060 GPU, a speedup of 32 % (35%) is obtained for Strassen’s 4level implementation and 33 % (36%) for Winograd’s variant relative to the sgemm (inte ..."
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Cited by 1 (0 self)
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Abstract—We provide efficient singleprecision and integer GPU implementations of Strassen’s algorithm as well as of Winograd’s variant. On an NVIDIA C1060 GPU, a speedup of 32 % (35%) is obtained for Strassen’s 4level implementation and 33 % (36%) for Winograd’s variant relative to the sgemm (integer version of sgemm) code in CUBLAS 3.0 when multiplying 16384×16384 matrices. The maximum numerical error for the singleprecision implementations is about 2 orders of magnitude higher than those for sgemm when n = 16384 and is zero for the integer implementations. KeywordsGPU; CUDA; matrix multiplication; Strassen’s algorithm; Winograd’s variant; accuracy; I.
Geometric Complexity Theory, Tensor Rank, and LittlewoodRichardson Coefficients
"... We provide a thorough introduction to Geometric Complexity Theory, an approach towards computational complexity lower bounds via methods from algebraic geometry and representation theory. Then we focus on the relevant representation theoretic multiplicities, namely plethysm coefficients, Kronecker c ..."
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We provide a thorough introduction to Geometric Complexity Theory, an approach towards computational complexity lower bounds via methods from algebraic geometry and representation theory. Then we focus on the relevant representation theoretic multiplicities, namely plethysm coefficients, Kronecker coefficients, and LittlewoodRichardson coefficients. These multiplicities can be described as dimensions of highest weight vector spaces for which explicit bases are known only in the LittlewoodRichardson case. By explicit construction of highest weight vectors we can show that the border rank of
Improving Numerical Accuracy for NonNegative Matrix Multiplication on GPUs using Recursive Algorithms
"... Scientific computing is only bound by the limits of Moore’s Law and the scalability of high performance mathematical library implementations. Most mathematical libraries however tend to focus only on general inputs, limiting their potential performance and scalability by not tailoring their implemen ..."
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Scientific computing is only bound by the limits of Moore’s Law and the scalability of high performance mathematical library implementations. Most mathematical libraries however tend to focus only on general inputs, limiting their potential performance and scalability by not tailoring their implementation to specific inputs, such as nonnegative inputs. By removing this limitation it is possible to improve the performance and accuracy of a range of problems. In this paper we explore the limitations of hardware to improve accuracy of nonnegative matrix multiply by specifically comparing implementations on the GPU and CPU and propose algorithmic solutions to improve accuracy. Next, we demonstrate a matrix multiply implementation that takes advantage of asymptotically fast matrix multiply algorithms, which have been shown to scale better than O(N 3) matrix multiply implementations, and improve accuracy by up to a whole digit while increasing performance by up to 27 % for matrices where the input is positive. Finally, we propose to extend the BLAS level 3 specification to nonnegative matrices to allow easy integration of our solution and allow other library authors to implement their own solutions as part of an existing standard.