Group-Theoretic Lower Bounds for the Complexity of Matrix Multiplication

摘要

The complexity of multiplication in group algebras is closely related to the complexity of matrix multiplication. Inspired by the recent group-theoretic approach by Cohn and Umans [10] and the algorithms by Cohn et al. [9] for matrix multiplication, we present conditional group-theoretic lower bounds for the complexity of matrix multiplication. These bounds depend on the complexity of multiplication in group algebras. Using Blaser's lower bounds for the rank of associative algebras we characterize all semisimple group algebras of minimal bilinear complex ity and show improved lower bounds for other group algebras. We also improve the best previously known bound for the bilinear complexity of group algebras by Atkinson. Our bounds depend on the complexity of matrix multiplication. In the special if the exponent of the matrix multiplication equals two, we achieve almost linear bounds.