Lower Bounds for Shallow Circuits

Prahladh Harsha
Monday, 23 Nov 2015, 16:00 to 17:00
A-212 (STCS Seminar Room)
Abstract: Over the last 3--4 years, there has been a sudden surge of results in arithmetic circuit lower bounds, specifically for shallow circuits (depth 3 or 4). The main motivation for studying depth-4 circuits is a result of Agrawal and Vinay (and subsequent strengthening by Koiran and Tavenas) that states that any arithmetic circuit of size s that computes an n-variate degree d polynomial can be simulated by a structured homogeneous depth-4 circuit of size n^{Omega (sqrt{d})}. Flipping this around, this implies that if we can find an explicit polynomial that requires such structured depth-4 circuits of size n^{omega(sqrt {d})}, then we would have proved lower bounds for general arithmetic circuits!

In 2013, in a joint work with Gupta, Kamath and Kayal, we showed a lower bound of 2^{Omega (sqrt{d})} for the dxd permanent or determinant using a technique called 'shifted partial derivatives' (introduced by [Kayal]). Subsequent to this result, there has been a sequence of improvements of various sorts (improving the lower bound, or dealing with a larger circuit class etc.) thereby improving our understanding of depth four circuits. The current state of the art is an n^{Omega(sqrt{d})} lower bound for the class of homogeneous depth-4circuits with no structural restrictions [Kayal-Limaye-Saha-Srinivasan, Kumar-Saraf].

In this talk, I shall give a brief overview of the progress made in the last 3 years and put the various results in context. Then, I shall present a recent work with Mrinal Kumar that gives an exponential lower bound for depth-5 circuits over small finite fields that combines the techniques developed so far with an old result of Grigoriev-Karpinski for depth-3 lower bounds.