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.2015.58
URN: urn:nbn:de:0030-drops-50600
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5060/
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Haviv, Ishay ; Regev, Oded

The List-Decoding Size of Fourier-Sparse Boolean Functions

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Abstract

A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most k nonzero Fourier coefficients. For a function f: F_2^n -> R and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n * k * log(k)).

As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz [Chicago J. Theor. Comput. Sci.,2013].

BibTeX - Entry

@InProceedings{haviv_et_al:LIPIcs:2015:5060,
  author =	{Ishay Haviv and Oded Regev},
  title =	{{The List-Decoding Size of Fourier-Sparse Boolean Functions}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{58--71},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{David Zuckerman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5060},
  URN =		{urn:nbn:de:0030-drops-50600},
  doi =		{10.4230/LIPIcs.CCC.2015.58},
  annote =	{Keywords: Fourier-sparse functions, list-decoding, learning theory, property testing}
}

Keywords: Fourier-sparse functions, list-decoding, learning theory, property testing
Collection: 30th Conference on Computational Complexity (CCC 2015)
Issue Date: 2015
Date of publication: 06.06.2015

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