On the Entropy of a Noisy Function

Let 0 < ϵ < 1/2 be a noise parameter, and let T_ϵ be the noise operator acting on functions on the Boolean cube {0,1}ⁿ. Let f be a nonnegative function on {0,1}ⁿ. We upper bound the entropy of T_ϵ f by the average entropy of conditional expectations of f, given sets of roughly (1-2 ϵ )². n variables. In information-theoretic terms, we prove the following strengthening of Mrs. Gerber's Lemma: let X be a random binary vector of length n, and let Z be a noise vector, corresponding to a binary symmetric channel with crossover probability ϵ . Then, setting v = (1-2 ϵ)². n, we have (up to lower order terms): H (X ⊕ Z) ≥ n H₂ ( ϵ + (1-2 ϵ) H₂^-1 (E_|B| = v H (X_i{i ∈ B)/v). Assuming ϵ ≥ 1/2-δ, for some absolute constant δ > 0, this inequality, combined with a strong version of a theorem of Friedgut et al., due to Jendrej et al., shows that if a Boolean function f is close to a characteristic function g of a subcube of dimension n-1, then the entropy of T_ϵ f is at most that of T_ϵ g. Taken together with a recent result of Ordentlich et al., this shows that the most informative Boolean function conjecture of Courtade and Kumar holds for high noise ϵ ≥ 1/2-δ. Namely, if X is uniformly distributed in {0,1ⁿ} and Y is obtained by flipping each coordinate of X independently with probability ϵ, then, provided ϵ ≥ 1/2-δ, for any Boolean function f holds I (f(X);Y ) ≤ 1-H(ϵ).