Fourier Concentration from Shrinkage

Abstract

For Boolean functions computed by de Morgan formulas of sub quadratic size or read-once de Morgan formulas, we prove a sharp concentration of the Fourier mass on small-degree coefficients. For a Boolean function f: {0, 1}^nto {1, -1} computable by a de Morgan formula of size s, we show that [ sum_{Asubseteq [n]:, |A|>s^{1/Gamma + epsilon}} hat{f}(A)^2 leq exp(-s^{epsilon/3}), ] where Gamma is the shrinkage exponent for the corresponding class of formulas: Gamma=2 for de Morgan formulas, and Gamma=1/log_2(sqrt{5}-1)approx 3.27 for read-once de Morgan formulas. We prove that this Fourier concentration is essentially optimal. As an application, we get that sub quadratic-size de Morgan formulas have negligible correlation with parity, and are learnable under the uniform distribution, and also lossily compressible, in sub exponential time. Finally, we establish the tight Theta(s^{1/Gamma}) bound on the average sensitivity of read-once formulas of size s, this mirrors the known tight bound Theta(sqrt{s}) on the average sensitivity of general de Morgan formulas of size s.