The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96]
seeks to relate two fundamental measures of Boolean function complexity: it
states that H[f]≤CInf[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 [OWZ11] 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