Electronic Journal of Probability

Noise stability and correlation with half spaces

Elchanan Mossel and Joe Neeman

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Benjamini, Kalai and Schramm showed that a monotone function $f : \{-1,1\}^n \to \{-1,1\}$ is noise stable if and only if it is correlated with a half-space (a set of the form $\{x: \langle x, a \rangle \le b\}$).

We study noise stability in terms of correlation with half-spaces for general (not necessarily monotone) functions. We show that a function $f: \{-1, 1\}^n \to \{-1, 1\}$ is noise stable if and only if it becomes correlated with a half-space when we modify $f$ by randomly restricting a constant fraction of its coordinates.

Looking at random restrictions is necessary: we construct noise stable functions whose correlation with any half-space is $o(1)$. The examples further satisfy that different restrictions are correlated with different half-spaces: for any fixed half-space, the probability that a random restriction is correlated with it goes to zero.

We also provide quantitative versions of the above statements, and versions that apply for the Gaussian measure on $\mathbb{R} ^n$ instead of the discrete cube. Our work is motivated by questions in learning theory and a recent question of Khot and Moshkovitz.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 16, 17 pp.

Received: 25 April 2017
Accepted: 25 January 2018
First available in Project Euclid: 23 February 2018

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Zentralblatt MATH identifier

Primary: 60

boolean function noise stability functional inequality

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Mossel, Elchanan; Neeman, Joe. Noise stability and correlation with half spaces. Electron. J. Probab. 23 (2018), paper no. 16, 17 pp. doi:10.1214/18-EJP145. https://projecteuclid.org/euclid.ejp/1519354945

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