An Improved Upper Bound for the Most Informative Boolean Function Conjecture

TL;DR

This paper introduces a new upper bound on the mutual information between a Boolean function of a binary vector and its noisy observation, improving previous bounds for certain noise levels and supporting the conjecture by Courtade and Kumar.

Contribution

It derives a novel upper bound for the mutual information of balanced Boolean functions, surpassing previous bounds in specific noise regimes.

Findings

New upper bound holds for all balanced functions.

Improves upon previous bounds for crossover probabilities between 1/3 and 1/2.

Supports the Courtade-Kumar conjecture in certain cases.

Abstract

Suppose $X$ is a uniformly distributed $n$-dimensional binary vector and $Y$ is obtained by passing $X$ through a binary symmetric channel with crossover probability $\alpha$. A recent conjecture by Courtade and Kumar postulates that $I(f(X);Y)\leq 1-h(\alpha)$ for any Boolean function $f$. So far, the best known upper bound was $I(f(X);Y)\leq (1-2\alpha)^2$. In this paper, we derive a new upper bound that holds for all balanced functions, and improves upon the best known bound for all $\tfrac{1}{3}<\alpha<\tfrac{1}{2}$.