License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2021.24
URN: urn:nbn:de:0030-drops-142988
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14298/
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Iyer, Vishnu ; Tal, Avishay ; Whitmeyer, Michael

Junta Distance Approximation with Sub-Exponential Queries

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LIPIcs-CCC-2021-24.pdf (1 MB)

Abstract

Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function f:{±1}ⁿ → {±1}:
1) We give a poly(k, 1/(ε)) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k'-juntas, where k' = O(k/(ε²)).
2) In the non-relaxed setting, we extend our ideas to give a 2^{Õ(√{k/ε})} (adaptive) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k-juntas. To the best of our knowledge, this is the first subexponential-in-k query algorithm for approximating the distance of f to being a k-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in k). Our techniques are Fourier analytical and make use of the notion of "normalized influences" that was introduced by Talagrand [Michel Talagrand, 1994].

BibTeX - Entry

@InProceedings{iyer_et_al:LIPIcs.CCC.2021.24,
  author =	{Iyer, Vishnu and Tal, Avishay and Whitmeyer, Michael},
  title =	{{Junta Distance Approximation with Sub-Exponential Queries}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{24:1--24:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14298},
  URN =		{urn:nbn:de:0030-drops-142988},
  doi =		{10.4230/LIPIcs.CCC.2021.24},
  annote =	{Keywords: Algorithms, Complexity Theory, Fourier Analysis, Juntas, Normalized Influence, Property Testing, Tolerant Property Testing}
}

Keywords: Algorithms, Complexity Theory, Fourier Analysis, Juntas, Normalized Influence, Property Testing, Tolerant Property Testing
Collection: 36th Computational Complexity Conference (CCC 2021)
Issue Date: 2021
Date of publication: 08.07.2021

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