Decision trees, protocols and the entropy-influence conjecture
Andrew Wan, John Wright, Chenggang Wu · 2014
Given ƒ : {--1, 1}n → {-- 1, 1}, define the spectral distribution of ƒ to be the distribution on subsets of [n] in which the set S is sampled with probability ƒ(S)2. Then the Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [2] states that there is some absolute constant C such that H[ƒ2] ≤ C ⋅ Inf[ƒ]. Here, H[ƒ2] denotes the Shannon entropy of ƒ's spectral distribution, and Inf[ƒ] is the total influence of ƒ. This conjecture is one of the major open problems in the analysis of Boolean functions, and settling it would have several interesting consequences.