Improved Low-Depth Set-Multilinear Circuit Lower Bounds

TL;DR

This paper establishes stronger lower bounds for constant-depth set-multilinear formulas computing explicit polynomials, advancing the understanding of algebraic formula complexity and surpassing previous bounds for specific models.

Contribution

It introduces improved lower bounds for set-multilinear formulas of constant depth, especially for NW design-based polynomials, and extends the techniques to general set-multilinear formulas.

Findings

Lower bounds of n^{Ω(n^{1/Δ}/Δ)} for depth-Δ formulas computing specific polynomials.

Improved lower bounds of n^{Ω(log n)} for general set-multilinear formulas computing NW polynomials.

New proof techniques that extend previous methods and yield stronger bounds.

Abstract

We prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial $f$ in VNP defined over $n^2$ variables, and of degree $n$, such that any product-depth $\Delta$ set-multilinear formula computing $f$ has size at least $n^{\Omega \left( n^{1/\Delta}/\Delta\right)}$. The hard polynomial $f$ comes from the class of Nisan-Wigderson (NW) design-based polynomials. Our lower bounds improve upon the recent work of Limaye, Srinivasan and Tavenas (STOC 2022), where a lower bound of the form $(\log n)^{\Omega (\Delta n^{1/\Delta})}$ was shown for the size of product-depth $\Delta$ set-multilinear formulas computing the iterated matrix multiplication (IMM) polynomial of the same degree and over the same number of variables as $f$. Moreover, our lower bounds are novel for any $\Delta\geq 2$. For general set-multilinear formulas, a lower bound of the form $ n^{\Omega(\log n)}$ was already obtained by Raz (J. ACM 2009) for the more general model of multilinear formulas. The techniques of LST give a different route to set-multilinear formula lower bounds, and allow them to obtain a lower bound of the form $(\log n)^{\Omega(\log n)}$ for the size of general set-multilinear formulas computing the IMM polynomial. Our proof techniques are another variation on those of LST, and enable us to show an improved lower bound (matching that of Raz) of the form $n^{\Omega(\log n)}$, albeit for the same polynomial $f$ in VNP (the NW polynomial). As observed by LST, if the same $n^{\Omega(\log n)}$ size lower bounds for unbounded-depth set-multilinear formulas could be obtained for the IMM polynomial, then using the self-reducibility of IMM and using hardness escalation results, this would imply super-polynomial lower bounds for general algebraic formulas.