A Composition Theorem for the Fourier Entropy-Influence Conjecture

The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [1] seeks to relate two fundamental measures of Boolean function complexity: it states that H[f]≤C· Inf[f] holds for every Boolean function f, where H[f] denotes the spectral entropy of f, Inf[f] is its total influence, and C>0 is a universal constant. Despite significant interest in the conjecture it has only been shown to hold for a few classes of Boolean functions. Our main result is a composition theorem for the FEI conjecture. We show that if g1,…,gk are functions over disjoint sets of variables satisfying the conjecture, and if the Fourier transform of F taken with respect to the product distribution with biases E[g1],..., E[gk] satisfies the conjecture, then their composition F(g1(x1),…,gk(xk)) satisfies the conjecture. As an application we show that the FEI conjecture holds for read-once formulas over arbitrary gates of bounded arity, extending a recent result [2] which proved it for read-once decision trees. Our techniques also yield an explicit function with the largest known ratio of C≥6.278 between H[f] and Inf[f], improving on the previous lower bound of 4.615.