Group-Theoretic Methods for Bounding the Exponent of Matrix Multiplication
Sandeep Murthy
The (asymptotic) complexity of matrix multiplication (over the complex
field) is measured by a real parameter w > 0, called the exponent of
matrix multiplication (over the complex field), which is defined to be
the smallest real number w > 0 such that for an arbitrary degree of
precision > 0, two n by n complex matrices can be multiplied using an
algorithm using O(n^(w+\epsilon)) number of non-division arithmetical
operations. By the standard algorithm for multiplying two matrices,
the trivial lower and upper bounds for the exponent w are 2 and 3
respectively.
W. Strassen in 1969 obtained the first important result that w < 2.81
using his result that 2 by 2 matrix multiplication could be performed
using 7 multiplications, not 8, as in the standard algorithm. In 1984,
V. Pan improved this to 2.67, using a variant of Strassen's
approach. It has been conjectured that w = 2, but the best known
result is that w < 2.38, due to D. Coppersmith and S. Winograd. In all
these approaches, estimates for w depend on the number of main running
steps in their algorithms.
In a recent series of papers in 2003 and 2005, H. Cohn and C. Umans
put forward an entirely different approach using fairly elementary
methods involving finite groups, group algebras and their
representations. The author describes the group-theoretical framework
behind their approach, proves their main results, and suggests
possible ways of getting improved estimates for the exponent using
their methods using wreath products of Abelian groups with symmetric
groups.