Upper Bounds on Fourier Entropy

Given a function f : {0, 1} n → R, its Fourier Entropy is de ned to be −∑S f̂(S) log f̂(S), where f̂ denotes the Fourier transform of f. This quantity arises in a number of applications, especially in the study of Boolean functions. An outstanding open question is a conjecture of Friedgut and Kalai (1996), called Fourier Entropy In uence (FEI) Conjecture, asserting that the Fourier Entropy of any Boolean function f is bounded above, up to a constant factor, by the total in uence (= average sensitivity) of f. In this paper we give several upper bounds on the Fourier Entropy of Boolean as well as real valued functions. We give a general bound involving the (1 + δ)-th moment of |S| w.r.t. the distribution f̂(S); the FEI conjecture needs the rst moment of |S|. A variant of this bound uses the rst and second moments of sensitivities (average sensitivity being the rst moment). We also give upper bounds on the Fourier Entropy of Boolean functions in terms of several complexity measures that are known to be bigger than the in uence. These complexity measures include, among others, the logarithm of the number of leaves and the average depth of a decision tree. Finally, we show that the FEI Conjecture holds for two special classes of functions, namely linear threshold functions and read-once formulas.