Toward an Optimal Algorithm for Matrix Multiplication

Ever since the dawn of the computer age, researchers have been trying to find an optimal way of multiplying matrices, a fundamental operation that is a bottleneck for many important algorithms. Faster matrix multiplication would give more efficient algorithms for many standard linear algebra problems, such as inverting matrices, solving systems of linear equations, and finding determinants. Even some basic graph algorithms run only as fast as matrix multiplication. The standard method for multiplying n × n matrices requires O(n3) multiplications. The fastest known algorithm, devised in 1987 by Don Coppersmith and Shmuel Winograd, runs in O(n2.38) time. Most researchers believe that an optimal algorithm will run in essentially O(n2) time, yet until recently no further progress was made in finding one. Two years ago, a theoretical computer scientist and a mathematician came up with a promising new group theoretic approach to fast matrix multiplication. The researchers—Chris Umans of the California Institute of Technology and Henry Cohn of Microsoft Research—showed that if there are groups that simultaneously satisfy two conditions, then the group theoretic approach will yield nontrivial (< 3) upper bounds on the exponent for matrix multiplication. Now, Umans and Cohn—together with Robert Kleinberg of the University of California at Berkeley and Balazs Szegedy of the Institute for Advanced Study in Princeton—have found a group that satisfies the two conditions. In a recent paper, the four collaborators describe a series of new matrix multiplication algorithms, the most efficient of which matches the time of the fastest known algorithm. The researchers also came up with two conjectures; a positive resolution to either would imply that the exponent for matrix multiplication is 2. “Our approach makes evident a connection between fast matrix multiplication and group theory, which potentially gives us more tools to work with,” Kleinberg says. The researchers presented their results in October at the IEEE Symposium on the Foundations of Computer Science, in Pittsburgh.