Abstract
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 g 1,…,g k 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[g 1],..., E[g k ] satisfies the conjecture, then their composition F(g 1(x 1),…,g k (x k)) 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.
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O’Donnell, R., Tan, LY. (2013). A Composition Theorem for the Fourier Entropy-Influence Conjecture. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_66
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