New lower bounds for the border rank of matrix multiplication

TL;DR

This paper establishes improved lower bounds on the border rank of matrix multiplication using algebraic geometry and representation theory, advancing understanding of its computational complexity.

Contribution

It introduces new equations for secant varieties of triple Segre products, leading to tighter lower bounds on matrix multiplication border rank.

Findings

Border rank is at least 2n^2 - n for n x n matrices.

Improved lower bounds surpass previous results for all n > 2.

New algebraic equations characterize border rank constraints.

Abstract

The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n^2-n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3/2 n^2+ n/2 -1 for all n>2. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.