eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-08-01
31:1
31:30
10.4230/LIPIcs.CCC.2017.31
article
An Exponential Lower Bound for Homogeneous Depth-5 Circuits over Finite Fields
Kumar, Mrinal
Saptharishi, Ramprasad
In this paper, we show exponential lower bounds for the class of homogeneous depth-5 circuits over all small finite fields. More formally, we show that there is an explicit family {P_d} of polynomials in VNP, where P_d is of degree d in n = d^{O(1)} variables, such that over all finite fields GF(q), any homogeneous depth-5 circuit which computes P_d must have size at least exp(Omega_q(sqrt{d})).
To the best of our knowledge, this is the first super-polynomial lower bound for this class for any non-binary field.
Our proof builds up on the ideas developed on the way to proving lower bounds for homogeneous depth-4 circuits [Gupta et al., Fournier et al., Kayal et al., Kumar-Saraf] and for non-homogeneous depth-3 circuits over finite fields [Grigoriev-Karpinski, Grigoriev-Razborov].
Our key insight is to look at the space of shifted partial derivatives of a polynomial as a space of functions from GF(q)^n to GF(q) as opposed to looking at them as a space of formal polynomials and builds over a tighter analysis of the lower bound of Kumar and Saraf [Kumar-Saraf].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol079-ccc2017/LIPIcs.CCC.2017.31/LIPIcs.CCC.2017.31.pdf
arithmetic circuits
lower bounds
separations
depth reduction