On read-once threshold formulae and their randomized decision tree complexity
Theoretical Computer Science - Special issue on structure in complexity theory
On learning monotone DNF under product distributions
Information and Computation
Learning Monotone Decision Trees in Polynomial Time
SIAM Journal on Computing
Agnostically learning decision trees
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Improved pseudorandom generators for depth 2 circuits
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
The Fourier entropy-influence conjecture for certain classes of Boolean functions
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Decision trees, protocols and the entropy-influence conjecture
Proceedings of the 5th conference on Innovations in theoretical computer science
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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 C0 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.