Decision trees, protocols and the entropy-influence conjecture

Abstract

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. Previous results on the FEI conjecture have been largely through direct calculation. In this paper we study a natural interpretation of the conjecture, which states that there exists a communication protocol which, given subset S of [n] distributed as ƒ2, can communicate the value of S using at most C⋅Inf[ƒ] bits in expectation. Using this interpretation, we are able show the following results: First, if ƒ is computable by a read-k decision tree, then H[ƒ2] ≤ 9k ⋅ Inf[ƒ]. Next, if ƒ has Inf[ƒ] ≥ 1 and is computable by a decision tree with expected depth d, then H[[ƒ2] ≤ 12d⋅ Inf[ƒ]. Finally, we give a new proof of the main theorem of O'Donnell and Tan [8], i.e. that their FEI+ conjecture composes. In addition, we show that natural improvements to our decision tree results would be sufficient to prove the FEI conjecture in its entirety. We believe that our methods give more illuminating proofs than previous results about the FEI conjecture.