Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Moshkovitz, Dana; Oh, Justin; Zuckerman, David https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-132721
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Randomness Efficient Noise Stability and Generalized Small Bias Sets

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Abstract

We present a randomness efficient version of the linear noise operator T_ρ from boolean function analysis by constructing a sparse linear operator on the space of boolean functions {0,1}ⁿ → {0,1} with similar eigenvalue profile to T_ρ. The linear operator we construct is a direct consequence of a generalization of ε-biased sets to the product distribution 𝒟_p on {0,1}ⁿ where the marginal of each coordinate is p = 1/2-1/2ρ. Such a generalization is a small support distribution that fools linear tests when the input of the test comes from 𝒟_p instead of the uniform distribution. We give an explicit construction of such a distribution that requires log n + O_{p}(log log n + log1/(ε)) bits of uniform randomness to sample from, where the p subscript hides O(log² 1/p) factors. When p and ε are constant, this yields a support size nearly linear in n, whereas previous best known constructions only guarantee a size of poly(n). Furthermore, our construction implies an explicitly constructible "sparse" noisy hypercube graph that is a small set expander.

BibTeX - Entry

@InProceedings{moshkovitz_et_al:LIPIcs:2020:13272,
  author =	{Dana Moshkovitz and Justin Oh and David Zuckerman},
  title =	{{Randomness Efficient Noise Stability and Generalized Small Bias Sets}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{31:1--31:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Nitin Saxena and Sunil Simon},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/13272},
  URN =		{urn:nbn:de:0030-drops-132721},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.31},
  annote =	{Keywords: pseudorandomness, derandomization, epsilon biased sets, noise stability}
}

Keywords: pseudorandomness, derandomization, epsilon biased sets, noise stability
Seminar: 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)
Issue date: 2020
Date of publication: 04.12.2020


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