Boolean functions: noise stability, non-interactive correlation distillation, and mutual information (1801.04462v6)

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.