Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits

Abstract: In this paper, we prove superpolynomial lower bounds for the class of homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP of degree $n$ in $n^2$ variables such that any homogeneous depth 4 arithmetic circuit computing it must have size $n^{\Omega(\log \log n)}$. Our results extend the works of Nisan-Wigderson NW95, Gupta-Kamath-Kayal-Saptharishi and Kayal-Saha-Saptharishi GKKS13, KSS13, Kumar-Saraf KS13a and Raz-Yehudayoff and Fournier-Limaye-Malod-Srinivasan RY08, FLMS13. Several of these results in fact showed exponential lower bounds. The main ingredient in our proof is a new complexity measure of {\it bounded support} shifted partial derivatives. This measure allows us to prove exponential lower bounds for homogeneous depth 4 circuits where all the monomials computed at the bottom layer have {\it bounded support} (but possibly unbounded degree/fan-in), strengthening the results of Gupta et al and Kayal et al [GKKS13, KSS13]. This new lower bound combined with a careful "random restriction" procedure (that transforms general depth 4 homogeneous circuits to depth 4 circuits with bounded support) gives us our final result.