Group-theoretic Algorithms for Matrix Multiplication

We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2. 41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.

Authors

• Henry Cohn
• Robert Kleinberg
• Balázs Szegedy
• Chris Umans

Keywords

• Computer science
• Standards development
• Linear algebra
• Matrices
• Upper bound
• Mathematics
• Fourier transforms
• Organizing
• Matrix Multiplication
• Conjecture
• Family Groups
• Running Time
• Properties Of Products
• Abelian Group
• Cyclic Group
• Finite Group
• Subset Of Group
• Symmetric Group
• Hypergraph
• Group Algebra
• Non-abelian Group
• Semidirect Product