Designing Strassen's algorithm for matrix multiplication

Abstract: In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than $O(n^3)$. While the recursive construction in his algorithm is very clear, the key gain was made by showing that $2 \times 2$ matrix multiplication could be performed with only $7$ multiplications instead of $8$. The latter construction was arrived at by a process of elimination and appears to come out of thin air. We give a transparent proof of this algorithm using only a simple unitary $2$-design (coming from an irreducible representation of a finite group) and a few easy lines of calculation. Moreover, using basic facts from the representation theory of finite groups, we use $2$-designs coming from group orbits to generalize our construction to all $n$ (although the resulting algorithms aren't optimal for $n \ge 3$).

The talk will include background on matrix multiplication, and we'll be able to give the full proof. Based on joint work with C. Moore.

computational complexitydiscrete mathematicsMathematics

Audience: researchers in the discipline

( paper | slides | video )

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LA Combinatorics and Complexity Seminar

Series comments: Password is on the seminar page. www.math.ucla.edu/~pak/seminars/CCSem-Fall-2020.htm

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