Low-depth arithmetic circuit lower bounds via shifted partials

TL;DR

This paper establishes super-polynomial lower bounds for low-depth arithmetic circuits using shifted partials measures, directly bounding homogeneous formulas without reduction to set-multilinear circuits, and applies these bounds to key polynomials.

Contribution

It introduces a novel approach to lower bounds for low-depth circuits using shifted partials, avoiding circuit conversion and random restrictions, and extends bounds to UPT formulas.

Findings

Super-polynomial lower bounds for iterated matrix multiplication

Super-polynomial lower bounds for Nisan-Wigderson polynomials

Bounds established for homogeneous and UPT formulas

Abstract

We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving lower bounds for low-depth set-multilinear circuits. An interesting aspect of our proof is that it does not require conversion of a circuit to a set-multilinear circuit, nor does it involve a random restriction. We are able to upper bound the measures for homogeneous formulas directly, without going via set-multilinearity. Our lower bounds hold for the iterated matrix multiplication as well as the Nisan-Wigderson design polynomials. We also define a subclass of homogeneous formulas which we call unique parse tree (UPT) formulas, and prove superpolynomial lower bounds for these. This generalizes the superpolynomial lower bounds for regular formulas in [Kayal-Saha-Saptharishi, STOC 2014], [Fournier-Limaye-Malod-Srinivasan, STOC 2014].