The maximum mutual information between the output of a discrete symmetric channel and several classes of Boolean functions of its input

TL;DR

This paper proves the Courtade-Kumar conjecture, establishing an upper bound on the mutual information between Boolean functions of a binary symmetric channel's input and output, for various classes of functions and all error probabilities.

Contribution

The paper provides a proof of the Courtade-Kumar conjecture for multiple classes of Boolean functions over all error probabilities in binary symmetric channels.

Findings

Mutual information is bounded by 1 - H(p) for the considered classes.

The proof applies Karamata's theorem and probability theory techniques.

The result holds for all n ≥ 2 and 0 ≤ p ≤ 1/2.

Abstract

We prove the Courtade-Kumar conjecture, for several classes of n-dimensional Boolean functions, for all $n \geq 2$ and for all values of the error probability of the binary symmetric channel, $0 \leq p \leq 1/2$. This conjecture states that the mutual information between any Boolean function of an n-dimensional vector of independent and identically distributed inputs to a memoryless binary symmetric channel and the corresponding vector of outputs is upper-bounded by $1-\operatorname{H}(p)$, where $\operatorname{H}(p)$ represents the binary entropy function. That is, let $\mathbf{X}=[X_1 \ldots X_n]$ be a vector of independent and identically distributed Bernoulli(1/2) random variables, which are the input to a memoryless binary symmetric channel, with the error probability in the interval $0 \leq p \leq 1/2$ and $\mathbf{Y}=[Y_1 \ldots Y_n]$ the corresponding output. Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be an n-dimensional Boolean function. Then, $\operatorname{MI}(f(X),Y) \leq 1-\operatorname{H}(p)$. Our proof employs Karamata's theorem, concepts from probability theory, transformations of random variables and vectors and algebraic manipulations.