Boolean functions: noise stability, non-interactive correlation distillation, and mutual information

TL;DR

This paper investigates extremal properties of Boolean functions under noise, identifying functions that maximize certain moments related to noise stability, with implications for correlation distillation and information theory.

Contribution

It characterizes maximizers of the $ ext{E}(T_ f)^$ moments in various noise regimes and extends results to more general settings like discrete tori and tree models.

Findings

Identifies extremal Boolean functions for low and high noise levels.

Establishes results for large moments  in various noise regimes.

Extends analysis to functions on discrete tori and tree models.

Abstract

Let $T_{\epsilon}$ be the noise operator acting on Boolean functions $f:\{0, 1\}^n\to \{0, 1\}$, where $\epsilon\in[0, 1/2]$ is the noise parameter. Given $\alpha>1$ and fixed mean $\mathbb{E} f$, which Boolean function $f$ has the largest $\alpha$-th moment $\mathbb{E}(T_\epsilon f)^\alpha$? This question has close connections with noise stability of Boolean functions, the problem of non-interactive correlation distillation, and Courtade-Kumar's conjecture on the most informative Boolean function. In this paper, we characterize maximizers in some extremal settings, such as low noise ($\epsilon=\epsilon(n)$ is close to 0), high noise ($\epsilon=\epsilon(n)$ is close to 1/2), as well as when $\alpha=\alpha(n)$ is large. Analogous results are also established in more general contexts, such as Boolean functions defined on discrete torus $(\mathbb{Z}/p\mathbb{Z})^n$ and the problem of noise stability in a tree model.