An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

Abstract

We show here a $2^{\Omega(\sqrt{d} \cdot \log N)}$ size lower bound for homogeneous depth four arithmetic formulas over fields of characteristic zero. That is, we give an explicit family of polynomials of degree $d$ on $N$ variables (with $N = d^3$ in our case) with 0, 1-coefficients such that for any representation of a polynomial $f$ in this family of the form $ f = \sum_{i} \prod_{j} Q_{ij}, $ where the $Q_{ij}$'s are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that $ \sum_{i, j} (\text{number of monomials of~} Q_{ij}) \geq 2^{\Omega (\sqrt{d} \cdot \log N)}. $ The abovementioned family, which we refer to as the Nisan--Wigderson design-based family of polynomials, is in the complexity class $\mathsf{VNP}$. Our work builds on recent lower bound results and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of [N. Kayal et al., in Symposium on Theory of Computing, ACM, New York, 2014, pp. 119--127] and the $N^{\Omega(\log \log N)}$ lower bound in the independent work of [M. Kumar and S. Saraf, in Automata, Languages, and Programming, Part I, Springer, Berlin, 2014, pp. 751--762].