Powers of Tensors and Fast Matrix Multiplication
Abstract
This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with existing approaches, this method is based on convex optimization, and thus has polynomialtime complexity. As an application, we use this method to study powers of the construction given by Coppersmith and Winograd [Journal of Symbolic Computation, 1990] and obtain the upper bound $\omega<2.3728639$ on the exponent of square matrix multiplication, which slightly improves the best known upper bound.
 Publication:

arXiv eprints
 Pub Date:
 January 2014
 arXiv:
 arXiv:1401.7714
 Bibcode:
 2014arXiv1401.7714L
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Computer Science  Symbolic Computation
 EPrint:
 28 pages