ABSTRACT
This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic construction 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 polynomial-time 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 ω < 2.3728639 on the exponent of square matrix multiplication, which slightly improves the best known upper bound.
- N. Alon, A. Shpilka, and C. Umans. On sunflowers and matrix multiplication. Computational Complexity, 22(2):219--243, 2013.Google ScholarCross Ref
- A. Ben-Tal and A. Nemirovski. Lectures on Modern Convex Optimization. SIAM, 2001. Google ScholarDigital Library
- M. Bläser. Fast Matrix Multiplication. Number 5 in Graduate Surveys. Theory of Computing Library, 2013.Google Scholar
- S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004. Google ScholarDigital Library
- P. Bürgisser, M. Clausen, and M. A. Shokrollahi. Algebraic complexity theory. Springer, 1997. Google ScholarDigital Library
- H. Cohn, R. D. Kleinberg, B. Szegedy, and C. Umans. Group-theoretic algorithms for matrix multiplication. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 379--388, 2005. Google ScholarDigital Library
- H. Cohn and C. Umans. Fast matrix multiplication using coherent configurations. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1074--1087, 2013. Google ScholarDigital Library
- D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9(3):251--280, 1990. Google ScholarDigital Library
- A. M. Davie and A. J. Stothers. Improved bound for complexity of matrix multiplication. Proceedings of the Royal Society of Edinburgh, 143A:351--370, 2013.Google ScholarCross Ref
- S.-C. Fang and J. H.-S. Tsao. Entropy optimization: interior point methods. In Encyclopedia of Optimization, pages 544--548. Springer, 2001.Google ScholarCross Ref
- Y. Filmus. Matrix multiplication I and II, 2014. Lecture notes available at http://www.cs.toronto.edu/~yuvalf/.Google Scholar
- F. Le Gall. Faster algorithms for rectangular matrix multiplication. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, pages 514--523, 2012. Google ScholarDigital Library
- A. Schönhage. Partial and total matrix multiplication. SIAM Journal on Computing, 10(3):434--455, 1981.Google ScholarCross Ref
- A. Stothers. On the Complexity of Matrix Multiplication. PhD thesis, University of Edinburgh, 2010.Google Scholar
- V. Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13:354--356, 1969.Google ScholarDigital Library
- V. Strassen. The asymptotic spectrum of tensors and the exponent of matrix multiplication. In Proceedings of the 27th Annual IEEE Symposium on Foundations of Computer Science, pages 49--54, 1986. Google ScholarDigital Library
- V. Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Proceedings of the 44th ACM Symposium on Theory of Computing, pages 887--898, 2012. Most recent version available at the author's homepage. Google ScholarDigital Library
Index Terms
- Powers of tensors and fast matrix multiplication
Recommendations
Fast Matrix Multiplication: Limitations of the Coppersmith-Winograd Method
STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of ComputingUntil a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time O(n2.3755). Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le~Gall has led to an improved algorithm ...
Fast sparse matrix multiplication
Let A and B two n×n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m0.7n1.2+n2+o(1)) algebraic operations (i.e., multiplications, additions ...
Geometric rank of tensors and subrank of matrix multiplication
CCC '20: Proceedings of the 35th Computational Complexity ConferenceMotivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. ...
Comments