License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2017.31
URN: urn:nbn:de:0030-drops-75142
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7514/
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Kumar, Mrinal ; Saptharishi, Ramprasad

An Exponential Lower Bound for Homogeneous Depth-5 Circuits over Finite Fields

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LIPIcs-CCC-2017-31.pdf (0.7 MB)

Abstract

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].

BibTeX - Entry

@InProceedings{kumar_et_al:LIPIcs:2017:7514,
  author =	{Mrinal Kumar and Ramprasad Saptharishi},
  title =	{{An Exponential Lower Bound for Homogeneous Depth-5 Circuits over Finite Fields}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{31:1--31:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{Ryan O'Donnell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7514},
  URN =		{urn:nbn:de:0030-drops-75142},
  doi =		{10.4230/LIPIcs.CCC.2017.31},
  annote =	{Keywords: arithmetic circuits, lower bounds, separations, depth reduction}
}

Keywords: arithmetic circuits, lower bounds, separations, depth reduction
Collection: 32nd Computational Complexity Conference (CCC 2017)
Issue Date: 2017
Date of publication: 01.08.2017

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