Proof of the Most Informative Boolean Function Conjecture

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 α. Recently, Courtade and Kumar postulates that I(f(X);Y) ≤ 1−Hb(α) for any Boolean function f [1]. In this paper, we provide a proof of the correctness of this conjecture. I. PROBLEM STATEMENT Let X be an N -dimensional binary random vector uniformly distributed over S N , {0, 1} and Y be the output of passing each component of X through a binary symmetric channel with crossover probability α ≤ 1/2. The following was recently conjectured by Courtade and Kumar [1]. Conjecture 1: For any Boolean function f : {0, 1} → {0, 1} it holds that I(f(X );Y ) ≤ 1−Hb(α); , (1) where I(f(X );Y ) is the mutual information between f(X ) and Y and Hb(p) , −p log p− (1− p) log(1− p) is the binary entropy function. For a dictatorship function, f(X ) = Xn, the conjectured upper bound (1) is attained with equality. Therefore, the conjecture can be interpreted as postulating that dictatorship is the most “informative” Boolean function, i.e., it achieves the maximal I(f(X );Y ). II. PRELIMINARIES We begin with introducing the notation that will be followed in the paper. A. Notation The space of all binary N -tuples are denoted by S N = {0, 1} . The constrained Hamming weigth of a vector x is defined as ωn(x N ) , n ∑ k=1 xk, n = 1, . . . , N. (2) We omit the index whenever there is no constraint, i.e., for n = N , we simply use ω(x ). The notation W ∼ B(α,N) represents a generic binary random vector of length N with i.i.d. components {Wn}n=1 which are Bernoulli distributed with parameter α. Next, we define the odd/even decomposition of any set into its disjoint subsets. Definition 1 (Odd/Even Decomposition): Let F ⊆ S N be a set consisting of unique binary N -tuples. We define the even component of the set F as F N 0 , {x N ∈ F | xN = 0}, (3) that is, it consists of elements whose last entries are 0. By the same token, the odd component is defined as F N 1 , F N \ F 0 . (4) Any such indexing of a set with n = 0 or n = 1 is exclusively used to represent this decomposition.