Results 1  10
of
19
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, ..."
Abstract

Cited by 35 (5 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
(Show Context)
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
"... ..."
(Show Context)
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 ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
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...
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. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
. 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...
MULTIPLICATIVE COMPLEXITY OF DIRECT SUM OF QUADRATIC SYSTEMS
"... We consider the quadratic complexity of certain sets of quadratic forms. We study classes of direct sums of quadratic forms. For these classes of problems we show that the complexity of one direct sum is the sum of the complexities of the summands and that every minimal quadratic algorithm for compu ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We consider the quadratic complexity of certain sets of quadratic forms. We study classes of direct sums of quadratic forms. For these classes of problems we show that the complexity of one direct sum is the sum of the complexities of the summands and that every minimal quadratic algorithm for computing the direct sums is a directsum algorithm.
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
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.
Computational Complexity and Numerical Stability of Linear Problems
, 2009
"... We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the algebraic complexity theory as well as the general error ana ..."
Abstract
 Add to MetaCart
We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the algebraic complexity theory as well as the general error analysis for matrix multiplication and related problems. We emphasize the central role played by the matrix multiplication problem and discuss historical and modern approaches to its solution. 1 Computational complexity of linear problems In algebraic complexity theory one is often interested in the number of arithmetic operations required to perform a given computation. This is called the total (arithmetic) complexity of the computation. 1 Moreover, it is often appropriate to count only multiplications (and divisions), but not additions or multiplications by fixed scalars. These notions can be formalized [BCS97, Definition 4.7]. For now, let us invoke Notation. Let F be a field, A be a Falgebra, and ϕ ∈ A be a function. The total arithmetic