Matrix Multiplication, a Little Faster

Abstract

Strassen's algorithm (1969) was the first sub-cubic matrix multiplication algorithm. Winograd (1971) improved its complexity by a constant factor. Many asymptotic improvements followed. Unfortunately, most of them have done so at the cost of very large, often gigantic, hidden constants. Consequently, Strassen-Winograd's O(nlog27) algorithm often outperforms other matrix multiplication algorithms for all feasible matrix dimensions. The leading coefficient of Strassen-Winograd's algorithm was believed to be optimal for matrix multiplication algorithms with 2x2 base case, due to a lower bound of Probert (1976). Surprisingly, we obtain a faster matrix multiplication algorithm, with the same base case size and asymptotic complexity as Strassen-Winograd's algorithm, but with the coefficient reduced from 6 to 5. To this end, we extend Bodrato's (2010) method for matrix squaring, and transform matrices to an alternative basis. We prove a generalization of Probert's lower bound that holds under change of basis, showing that for matrix multiplication algorithms with a 2x2 base case, the leading coefficient of our algorithm cannot be further reduced, hence optimal. We apply our technique to other Strassen-like algorithms, improving their arithmetic and communication costs by significant constant factors.