## Electronic Journal of Probability

### Noise stability and correlation with half spaces

#### Abstract

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

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

Dates
Accepted: 25 January 2018
First available in Project Euclid: 23 February 2018

https://projecteuclid.org/euclid.ejp/1519354945

Digital Object Identifier
doi:10.1214/18-EJP145

Mathematical Reviews number (MathSciNet)
MR3771753

Zentralblatt MATH identifier
06868361

Subjects
Primary: 60

#### Citation

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

#### References

• [1] D. Bakry, I. Gentil, and M. Ledoux. Analysis and geometry of Markov diffusion operators, volume 348. Springer, 2014.
• [2] I. Benjamini, G. Kalai, and O. Schramm. Noise sensitivity of boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math., 90:5–43, 1999.
• [3] D. M. Kane. The Gaussian surface area and noise sensitivity of degree-d polynomial threshold functions. Computational Complexity, 20(2):389–412, 2011.
• [4] S. Khot and D. Moshkovitz. Candidate hard Unique Game. In STOC, 2016. to appear.
• [5] A. Klivans, R. O’Donnell, and R. Servedio. Learning intersections and thresholds of halfspaces. Journal of Computer and System Sciences, 68(4):808–840, 2004.
• [6] M. Ledoux. Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space. Bulletin des sciences mathématiques, 118(6):485–510, 1994.
• [7] N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform and learnability. Journal of the ACM, 40(3):607–620, 1993.
• [8] E. Mossel. Gaussian bounds for noise correlation of functions. GAFA, 19:1713–1756, 2010.
• [9] R. O’Donnell. Analysis of Boolean functions. Cambridge University Press, 2014.
• [10] Y. Peres. Noise stability of weighted majority. arXiv:math/0412377, 2004.