A composition theorem for the Fourier Entropy-Influence conjecture [article]

Ryan O'Donnell, Li-Yang Tan
2013 arXiv   pre-print
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] ≤ 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
more » ... onjecture. 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 [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.
arXiv:1304.1347v1 fatcat:3rp3cwf5qzcqtclbu3dwmftp5u