Arithmetic Circuit Lower Bounds via MaxRank

TL;DR

This paper introduces a new complexity measure for multivariate polynomials and uses it to establish super-polynomial lower bounds for various classes of arithmetic circuits, advancing understanding of computational complexity.

Contribution

It develops the polynomial coefficient matrix and max-rank measure, leading to new lower bounds for depth-3 circuits, diagonal circuits, product-sparse formulas, and partitioned arithmetic branching programs.

Findings

Proves depth-3 circuit lower bounds for matrix product computations.

Establishes exponential lower bounds for certain explicit polynomials with restricted circuit dimensions.

Extends super-polynomial lower bounds to product-sparse formulas and partitioned arithmetic branching programs.

Abstract

We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results : As our main result, we prove that any homogeneous depth-3 circuit for computing the product of $d$ matrices of dimension $n \times n$ requires $\Omega(n^{d-1}/2^d)$ size. This improves the lower bounds by Nisan and Wigderson(1995) when $d=\omega(1)$. There is an explicit polynomial on $n$ variables and degree at most $\frac{n}{2}$ for which any depth-3 circuit $C$ of product dimension at most $\frac{n}{10}$ (dimension of the space of affine forms feeding into each product gate) requires size $2^{\Omega(n)}$. This generalizes the lower bounds against diagonal circuits proved by Saxena(2007). Diagonal circuits are of product dimension 1. We prove a $n^{\Omega(\log n)}$ lower bound on the size of product-sparse formulas. By definition, any multilinear formula is a product-sparse formula. Thus, our result extends the known super-polynomial lower bounds on the size of multilinear formulas by Raz(2006). We prove a $2^{\Omega(n)}$ lower bound on the size of partitioned arithmetic branching programs. This result extends the known exponential lower bound on the size of ordered arithmetic branching programs given by Jansen(2008).