Quantitative relation between noise sensitivity and influences

Nathan Keller, Guy Kindler

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

A Boolean function f: {0,1}n → {0,1} is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm [3] showed that if the sum of squares of inuences of f is close to zero then f must be noise sensitive. We show a quantitative version of this result which does not depend on n, and prove that it is tight for certain parameters. Our results hold also for a general product measure μp on the discrete cube, as long as log1/p≪logn. We note that in [3], a quantitative relation between the sum of squares of the inuences and the noise sensitivity was also shown, but only when the sum of squares is bounded by n -c for a constant c. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand's lemma, which easily generalizes in various directions, including non-monotone functions.

Original languageEnglish
Pages (from-to)45-71
Number of pages27
JournalCombinatorica
Volume33
Issue number1
DOIs
StatePublished - Feb 2013
Externally publishedYes

Bibliographical note

Funding Information:
∗ Partially supported by the Adams Fellowship Program of the Israeli Academy of Sciences and Humanities and by the Koshland Center for Basic Research. † Supported by the Israel Science Foundation and by the Binational Science Foundation.

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