Lower Bounds for Matrix Product

TL;DR

This paper establishes near-optimal lower bounds on the number of product gates needed in bilinear and quadratic circuits to multiply two matrices over finite fields, advancing previous bounds.

Contribution

It provides improved lower bounds for the number of product gates in matrix multiplication circuits over finite fields, refining earlier results from Bshouty and Blaser.

Findings

Lower bound of 3n^2 - o(n^2) over F_2

Lower bound of (2.5 + 1.5/(p^3 -1))n^2 - o(n^2) over F_p

Improved previous bounds from 2.5 n^2 - o(n^2)

Abstract

We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two $n \cross n$ matrices over finite fields. In particular we obtain the following results: 1. We show that the number of product gates in any bilinear (or quadratic) circuit that computes the product of two $n \cross n$ matrices over $F_2$ is at least $3 n^2 - o(n^2)$. 2. We show that the number of product gates in any bilinear circuit that computes the product of two $n \cross n$ matrices over $F_p$ is at least $(2.5 + \frac{1.5}{p^3 -1})n^2 -o(n^2)$. These results improve the former results of Bshouty '89 and Blaser '99 who proved lower bounds of $2.5 n^2 - o(n^2)$.