On the Most Informative Boolean Functions of the Very Noisy Channel

TL;DR

This paper investigates the maximum mutual information of Boolean functions transmitted over a noisy channel, proving the dictator function is most informative in high noise regimes using a calculus-based approach.

Contribution

It introduces a calculus-based method to analyze the conjecture on Boolean functions' informativeness, providing a dimension-dependent result and identifying the dictator function as optimal in high noise.

Findings

The dictator function is the most informative in high noise regimes.

A calculus-based approach yields dimension-dependent results.

The conjecture holds in the high noise regime, with new insights into Boolean functions.

Abstract

Let $X^n$ be a uniformly distributed $n$-dimensional binary vector, and $Y^n$ be the result of passing $X^n$ through a binary symmetric channel (BSC) with crossover probability $\alpha$. A recent conjecture postulated by Courtade and Kumar states that for any Boolean function $f:\{0,1\}^n\to\{0,1\}$, $I(f(X^n);Y^n)\le 1-H(\alpha)$. Although the conjecture has been proved to be true in the dimension-free high noise regime by Samorodnitsky, here we present a calculus-based approach to show a dimension-dependent result by examining the second derivative of $H(\alpha)-H(f(X^n)|Y^n)$ at $\alpha=1/2$. Along the way, we show that the dictator function is the most informative function in the high noise regime.