On Matrix Multiplication and Polynomial Identity Testing

TL;DR

This paper links lower bounds on matrix multiplication border rank to derandomizing polynomial identity testing, providing new hitting set generators with seed lengths dependent on matrix multiplication complexity.

Contribution

It introduces a novel approach connecting matrix multiplication lower bounds to polynomial identity testing derandomization, yielding new explicit hitting set generators.

Findings

Hitting set generator with seed length $O(\sqrt{n} imes ext{border rank}^{-1}(s))$

Generator hits circuits of size $O(n^{1+ ext{small }\delta})$

Border rank lower bounds imply non-trivial hitting sets for small circuits.

Abstract

We show that lower bounds on the border rank of matrix multiplication can be used to non-trivially derandomize polynomial identity testing for small algebraic circuits. Letting $\underline{R}(n)$ denote the border rank of $n \times n \times n$ matrix multiplication, we construct a hitting set generator with seed length $O(\sqrt{n} \cdot \underline{R}^{-1}(s))$ that hits $n$-variate circuits of multiplicative complexity $s$. If the matrix multiplication exponent $\omega$ is not 2, our generator has seed length $O(n^{1 - \varepsilon})$ and hits circuits of size $O(n^{1 + \delta})$ for sufficiently small $\varepsilon, \delta > 0$. Surprisingly, the fact that $\underline{R}(n) \ge n^2$ already yields new, non-trivial hitting set generators for circuits of sublinear multiplicative complexity.