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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 14 (2 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.
Fast Matrix Multiplication is Stable
, 2006
"... We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63–72]. As a consequence of our analysis, we show that the exponent of matrix mult ..."
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Cited by 10 (3 self)
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We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63–72]. As a consequence of our analysis, we show that the exponent of matrix multiplication can be achieved by numerically stable algorithms. We also show that new grouptheoretic algorithms proposed in [H. Cohn, and C. Umans, A grouptheoretic approach to fast matrix multiplication, FOCS 2003, 438–449] and [H. Cohn, R. Kleinberg, B. Szegedy and C. Umans, Grouptheoretic algorithms for matrix multiplication, FOCS 2005, 379–388] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast grouptheoretic algorithms. 1
Multiplicative complexity of polynomial multiplication over finite fields
 Proceedings 28th Annual Symposium on Foundations of Computer Science
, 1987
"... finite fields* ..."
Tensor Codes for the Rank Metric
 IEEE Trans. Inform. Theory
, 1995
"... Linear spaces of n \Theta n \Theta n tensors over finite fields are investigated where the rank of every nonzero tensor in the space is bounded from below by a prescribed number ¯. Such linear paces can recover any n \Theta n \Theta n error tensor of rank (¯\Gamma1)=2, and, as such, they can be us ..."
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Cited by 3 (1 self)
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Linear spaces of n \Theta n \Theta n tensors over finite fields are investigated where the rank of every nonzero tensor in the space is bounded from below by a prescribed number ¯. Such linear paces can recover any n \Theta n \Theta n error tensor of rank (¯\Gamma1)=2, and, as such, they can be used to correct threeway crisscross errors. Bounds on the dimensions of such spaces are given for ¯ 2n+1, and constructions are provided for ¯ 2n\Gamma1 with redundancy which is linear in n. These constructions can be generalized to spaces of n \Theta n \Theta \Delta \Delta \Delta \Theta n hyperarrays. Keywords: Algebraic computation, Crisscross errors, Tensor rank. This work was presented in part at the IEEE International Symposium on Information Theory, Whistler, BC, Canada, September 1995. y This research was supported by the Fund for the Promotion of Research at the Technion and by the Technion V.P.R Steiner Research Fund. 1 Introduction An n \Theta n \Theta n tensor over a fiel...
Rank of 3tensors with 2 slices and Kronecker canonical forms, preprint
"... Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A + T is diagonalizable for a given 3tensor A with 2 slices over the complex and real number field. 1 ..."
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Cited by 2 (2 self)
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Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A + T is diagonalizable for a given 3tensor A with 2 slices over the complex and real number field. 1
Multiplication of Polynomials Over Finite Fields
 SIAM Journal on Computing
, 1990
"... . We prove the 2.5n  o (n ) lower bound on the number of multiplications/divisions required to compute the coefficients of the product of two polynomials of degree n over a finite field by means of straightline algorithms. 1. Introduction The number of multiplications/divisions required for comput ..."
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Cited by 1 (1 self)
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. We prove the 2.5n  o (n ) lower bound on the number of multiplications/divisions required to compute the coefficients of the product of two polynomials of degree n over a finite field by means of straightline algorithms. 1. Introduction The number of multiplications/divisions required for computing the product of two degreen polynomials over an infinite field is known to be 2n + 1. The method is to evaluate both polynomials at each of 2n + 1 distinct points (allowing ), multiplying and interpolating the result. This method fails for the fields with the number of elements less than 2n . The bilinear and quadratic complexity of polynomial multiplication over finite fields has been widely studied in the literature, cf. [BD1], [BD2], [J2], [KA], [KB2], [LSW] and [LW]. The best known lower bounds on the bilinear complexity of multiplying two degreen polynomials over finite fields are as follows. For the binary field the bound is 3.52n , cf. [BD2]. The same bound holds for the quadrati...
Algebras of Minimal Rank over Perfect Fields
 In Proc. 17th Ann. IEEE Computational Complexity Conf. (CCC
, 2002
"... Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the socalled Alder Strassen bound R(A) 2 dim A \Gamma t, where t is the number of maximal twosided ideals of A. The class of algebras for which the Alde ..."
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Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the socalled Alder Strassen bound R(A) 2 dim A \Gamma t, where t is the number of maximal twosided ideals of A. The class of algebras for which the AlderStrassen bound is sharp, the socalled algebras of minimal rank, has received a wide attention in algebraic complexity theory.
Maximal Rank of m × n × (mn  k) Tensors
"... It is shown that the maximal rank of m × n × ( m n  k ) tensors with k min {( m  1 ) 2 /2 , ( n  1 ) 2 /2} is greater than m n  4Ö ### 2 k + O ( 1 ) . 1. INTRODUCTION A classical problem in algebraic computational complexity is to determine the minimal number of nonscalar multiplications ..."
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It is shown that the maximal rank of m × n × ( m n  k ) tensors with k min {( m  1 ) 2 /2 , ( n  1 ) 2 /2} is greater than m n  4Ö ### 2 k + O ( 1 ) . 1. INTRODUCTION A classical problem in algebraic computational complexity is to determine the minimal number of nonscalar multiplications required to evaluate some set S i ,j a i ,j ,k x i y j , k = 1, . . . , p , of bilinear forms in noncommuting variables x 1 , . . . , x m and y 1 , . . . , y n over a field F . This number is equal to the rank of the defining 3tensor ( a i ,j ,k ) ÎF m ÄF n ÄF p , cf. [S]. An interesting problem, which does not depend on the coefficients a i ,j ,k , is the determination of R F ( m , n , p ) = max T ÎF m ÄF n ÄF p rank of T , the maximal rank of tensors in F m ÄF n ÄF p . This problem has been studied quite extensively in [AL1AS, GatJ]. Atkinson and Stephens [AS] gave the following general reduction of R F ( m , n , m n  k ) with k min { m , n } to R F ( k , k , k 2 ...
A 5/2 n² Lower Bound for the Rank of n×nMatrix Multiplication over Arbitrary Fields
 IN 40TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (NEW
, 1999
"... We prove a lower bound of 5/2 n²  3n for the rank of n×nmatrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices. ..."
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We prove a lower bound of 5/2 n²  3n for the rank of n×nmatrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.
A 5/2 n²Lower Bound for the Multiplicative Complexity of n×nMatrix Multiplication
, 2001
"... We prove a lower bound of \Gamma 3n for the multiplicative complexity of n × nmatrix multiplication over arbitrary fields. More general, we show that for any finite dimensional semisimple algebra A with unity, the multiplicative complexity of the multiplication in A is bounded from below by ..."
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We prove a lower bound of \Gamma 3n for the multiplicative complexity of n × nmatrix multiplication over arbitrary fields. More general, we show that for any finite dimensional semisimple algebra A with unity, the multiplicative complexity of the multiplication in A is bounded from below by 2 dimA \Gamma 3(n1 + \Delta \Delta \Delta + n t ) if the decomposition of A = A1 \Theta \Delta \Delta \Delta \Theta A t into simple algebras A = D contains only noncommutative factors, that is, the division algebra D is noncommutative or n ≥ 2.