Information Geometric Approach on Most Informative Boolean Function Conjecture

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 vec...

The "Most informative boolean function" conjecture holds for high noise

We prove the ”Most informative boolean function” conjecture of Courtade and Kumar for high noise ǫ ≥ 1/2− δ, for some absolute constant δ > 0. Namely, ifX is uniformly distributed in {0, 1}n and Y is obtained by flipping each coordinate of X independently with probability ǫ, then, provided ǫ ≥ 1/2− δ, for any boolean function f holds I ( f(X);Y ) ≤ 1 − H(ǫ). This conjecture was previously known...

Remarks on the Most Informative Function Conjecture at fixed mean

In 2013, Courtade and Kumar posed the following problem: Let x ∼ {±1}n be uniformly random, and form y ∼ {±1}n by negating each bit of x independently with probability α. Is it true that the mutual information I(f(x) ; y) is maximized among f : {±1}n → {±1} by f(x) = x1? We do not resolve this problem. Instead, we resolve the analogous problem in the settings of Gaussian space and the sphere. O...

Lexical stress and phonetic information: which segments are most informative?

Dictatorship is the Most Informative Balanced Function at the Extremes

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. A recent conjecture by Courtade and Kumar postulates that I(f(X);Y ) ≤ 1− h(α) for any Boolean function f . In this paper, we prove the conjecture for all α ∈ [0, αn], and under the restriction to balanced functions, also for all α ∈ [1...