Noise Stability and Correlation with Half Spaces

TL;DR

This paper extends the understanding of noise stability from monotone functions to general functions, showing that noise stability relates to correlation with half-spaces after random restrictions, with implications for learning theory.

Contribution

It demonstrates that general noise stable functions become correlated with half-spaces under random restrictions, and constructs counterexamples showing limitations of correlation without restrictions.

Findings

Noise stability characterized by correlation with half-spaces after restrictions

Existence of noise stable functions with low correlation to any fixed half-space

Quantitative and Gaussian measure versions of the main results

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.