License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ICALP.2016.37
URN: urn:nbn:de:0030-drops-63173
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6317/
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Bun, Mark ; Thaler, Justin

Improved Bounds on the Sign-Rank of AC^0

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LIPIcs-ICALP-2016-37.pdf (0.6 MB)

Abstract

The sign-rank of a matrix A with entries in {-1, +1} is the least rank of a real matrix B with A_{ij}*B_{ij} > 0 for all i, j. Razborov and Sherstov (2008) gave the first exponential lower bounds on the sign-rank of a function in AC^0, answering an old question of Babai, Frankl, and Simon (1986). Specifically, they exhibited a matrix A = [F(x,y)]_{x,y} for a specific function F:{-1,1}^n*{-1,1}^n -> {-1,1} in AC^0, such that A has sign-rank exp(Omega(n^{1/3}).

We prove a generalization of Razborov and Sherstov’s result, yielding exponential sign-rank lower bounds for a non-trivial class of functions (that includes the function used by Razborov and Sherstov). As a corollary of our general result, we improve Razborov and Sherstov's lower bound on the sign-rank of AC^0 from exp(Omega(n^{1/3})) to exp(~Omega(n^{2/5})). We also describe several applications to communication complexity, learning theory, and circuit complexity.

BibTeX - Entry

@InProceedings{bun_et_al:LIPIcs:2016:6317,
  author =	{Mark Bun and Justin Thaler},
  title =	{{Improved Bounds on the Sign-Rank of AC^0}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{37:1--37:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6317},
  URN =		{urn:nbn:de:0030-drops-63173},
  doi =		{10.4230/LIPIcs.ICALP.2016.37},
  annote =	{Keywords: Sign-rank, circuit complexity, communication complexity, constant-depth circuits}
}

Keywords: Sign-rank, circuit complexity, communication complexity, constant-depth circuits
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016

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