Noise sensitivity on the p-biased hypercube

Noam Lifshitz, Dor Minzer

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations

Abstract

The noise sensitivity of a Boolean function measures how susceptible the value of f on a typical input x to a slight perturbation of the bits of x: it is the probability f(x) and f(y) are different when x is a uniformly chosen n-bit Boolean string, and y is formed by flipping each bit of x with small probability ϵ. The noise sensitivity of a function is a key concept with applications to combinatorics, complexity theory, learning theory, percolation theory and more. In this paper, we investigate noise sensitivity on the p-biased hypercube, extending the theory for polynomially small p. Specifically, we give sufficient conditions for monotone functions with large groups of symmetries to be noise sensitive (which in some cases are also necessary). As an application, we show that the 2-SAT function is noise sensitive around its critical probability. En route, we study biased versions of the invariance principle for monotone functions and give p-biased versions of Bourgain's tail theorem and the Majority is Stablest theorem, showing that in this case the correct analog of ''small low degree influences'' is lack of correlation with constant width DNF formulas.

Original languageAmerican English
Title of host publicationProceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019
PublisherIEEE Computer Society
Pages1205-1226
Number of pages22
ISBN (Electronic)9781728149523
DOIs
StatePublished - Nov 2019
Event60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 - Baltimore, United States
Duration: 9 Nov 201912 Nov 2019

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2019-November
ISSN (Print)0272-5428

Conference

Conference60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019
Country/TerritoryUnited States
CityBaltimore
Period9/11/1912/11/19

Bibliographical note

Publisher Copyright:
© 2019 IEEE.

Keywords

  • Analysis of Boolean Functions
  • Graph Properties
  • Noise Sensitivity

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