The "Most informative boolean function" conjecture holds for high noise

TL;DR

This paper proves the

Contribution

It establishes the conjecture for high noise levels, extending prior results limited to balanced functions, thus advancing understanding of information theory in noisy boolean functions.

Findings

The conjecture holds for noise levels $\epsilon \geq 1/2 - \delta$.

Provides bounds on mutual information for boolean functions under high noise.

Extends the validity of the conjecture beyond balanced functions.

Abstract

We prove the "Most informative boolean function" conjecture of Courtade and Kumar for high noise $\epsilon \ge 1/2 - \delta$, for some absolute constant $\delta > 0$. Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $\epsilon$, then, provided $\epsilon \ge 1/2 - \delta$, for any boolean function $f$ holds $I(f(X);Y) \le 1 - H(\epsilon)$. This conjecture was previously known to hold only for balanced functions.