Quantitative relation between noise sensitivity and influences

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 µ p 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.

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Correspondence to Nathan Keller.

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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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Keller, N., Kindler, G. Quantitative relation between noise sensitivity and influences. Combinatorica 33, 45–71 (2013). https://doi.org/10.1007/s00493-013-2719-2

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Mathematics Subject Classification (2000)

  • 05D40
  • 60C05
  • 06E30