Proof of the Most Informative Boolean Function Conjecture

TL;DR

This paper proves Courtade and Kumar's conjecture that the mutual information between any Boolean function of a binary vector and its noisy version is bounded by a specific function of the noise level.

Contribution

The paper provides a rigorous proof confirming the conjecture that bounds mutual information for all Boolean functions under binary symmetric noise.

Findings

The conjecture by Courtade and Kumar is proven correct.

Mutual information is bounded by 1 minus the binary entropy of the noise probability.

The result applies universally to all Boolean functions under the specified noise model.

Abstract

Suppose $\XX{N}$ is a uniformly distributed $N$-dimensional binary vector and $\YY{N}$ is obtained by passing $\XX{N}$ through a binary symmetric channel with crossover probability $\alpha$. Recently, Courtade and Kumar postulates that $I(f(\XX{N});\YY{N})\leq 1-\Be(\alpha)$ for any Boolean function $f$ \cite{courtade}. In this paper, we provide a proof of the correctness of this conjecture.