On the Power of Homogeneous Depth 4 Arithmetic Circuits

We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in $VP$. Our results hold for the iterated matrix multiplication polynomial---in particular we show that any homogeneous depth 4 circuit computing the (1,1) entry in the product of $n$ generic matrices of dimension $n^{O(1)}$ must have size $n^{\Omega(\sqrt{n})}$. Our results strengthen previous works in two significant ways: (1) Our lower bounds hold for a polynomial in $VP$. Prior to our work, Kayal et al. [Proceedings of FOCS, 2014, pp. 61--70] proved an exponential lower bound for homogeneous depth 4 circuits (over fields of characteristic zero) computing a poly in $VNP$. The best known lower bounds for a depth 4 homogeneous circuit computing a poly in $VP$ was the bound of $n^{\Omega(\log n)}$ by Kayal et al. Our exponential lower bounds also give the first exponential separation between general arithmetic circuits and homogeneous depth 4 arithmetic circuits. In particular they im...