Matrix rigidity and the Croot-Lev-Pach lemma

TL;DR

This paper demonstrates that matrices of the form M(x,y)=f(x+y) over finite fields are not sufficiently rigid for Valiant's circuit lower bound approach, leveraging the Croot-Lev-Pach lemma related to the cap-set problem.

Contribution

It shows a broad non-rigidity result for a class of structured matrices using the Croot-Lev-Pach lemma, challenging their potential for proving circuit lower bounds.

Findings

Matrices of the form M(x,y)=f(x+y) are not sufficiently rigid.

The result applies to all such matrices over finite fields as dimension grows.

The proof relies on the Croot-Lev-Pach lemma used in cap-set problem solutions.

Abstract

Matrix rigidity is a notion put forth by Valiant as a means for proving arithmetic circuit lower bounds. A matrix is rigid if it is far, in Hamming distance, from any low rank matrix. Despite decades of efforts, no explicit matrix rigid enough to carry out Valiant's plan has been found. Recently, Alman and Williams showed, contrary to common belief, that the $2^n \times 2^n$ Hadamard matrix could not be used for Valiant's program as it is not sufficiently rigid. In this note we observe a similar `non rigidity' phenomena for any $q^n \times q^n$ matrix $M$ of the form $M(x,y) = f(x+y)$, where $f:F_q^n \to F_q$ is any function and $F_q$ is a fixed finite field of $q$ elements ($n$ goes to infinity). The theorem follows almost immediately from a recent lemma of Croot, Lev and Pach which is also the main ingredient in the recent solution of the cap-set problem.