Algebraic Methods for Fast Matrix Multiplication

An improvement upon the naive O(n3) algorithm for matrix multiplication was first presented by Strassen, obtaining the result in only O(n2.81) field operations [5]. This raises the question of what the best possible exponent k such that matrix multiplication can be carried out in at most O(nk) time is. Clearly k ≥ 2, since n2 is the size of the output. It is believed that the optimal k is exactly equal to 2, but this has yet to be proven [3]. Presently, the fastest known algorithm computes the product of two matrices in at most O(n2.38) operations. Recent work by Cohn and Umans indicates a possible path to proving that the obvious lower bound of 2 is tight, namely an approach using techniques from group theory and representation theory [3]. Their proposed algorithm is analogous to the way that the Discrete Fourier Transform (DFT) computes the product of two polynomials by embedding them in a cyclic group algebra over the complex numbers, then computing the pointwise product of vectors in the appropriate complex vector space. Since abelian groups cannot yield the properties necessary to achieve k = 2, it is instead necessary to embed the matrices in a nonabelian group algebra. This complicates multiplication in the Fourier domain; rather than being simply pointwise vector multiplication, it becomes multiplication of block-diagonal matrices with block sizes determined by the irreducible representations of the group.