Improved bounds on Fourier entropy and Min-entropy
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Given a Boolean function f:{-1,1}^n→{-1,1}, the Fourier distribution assigns probability f(S)^2 to S⊆ [n]. The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai asks if there exist a universal constant C>0 such that H(f̂^2)≤ C Inf(f), where H(f̂^2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f. 1) We consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture. This asks if H_∞(f̂^2)≤ C Inf(f), where H_∞(f̂^2) is the min-entropy of the Fourier distribution. We show H_∞(f̂^2)≤ 2C_^⊕(f), where C_^⊕(f) is the minimum parity certificate complexity of f. We also show that for every ϵ≥ 0, we have H_∞(f̂^2)≤ 2 (f̂_1,ϵ/(1-ϵ)), where f̂_1,ϵ is the approximate spectral norm of f. As a corollary, we verify the FMEI conjecture for the class of read-k DNFs (for constant k). 2) We show that H(f̂^2)≤ 2 aUC^⊕(f), where aUC^⊕(f) is the average unambiguous parity certificate complexity of f. This improves upon Chakraborty et al. An important consequence of the FEI conjecture is the long-standing Mansour's conjecture. We show that a weaker version of FEI already implies Mansour's conjecture: is H(f̂^2)≤ C {C^0(f),C^1(f)}?, where C^0(f), C^1(f) are the 0- and 1-certificate complexities of f, respectively. 3) We study what FEI implies about the structure of polynomials that 1/3-approximate a Boolean function. We pose a conjecture (which is implied by FEI): no "flat" degree-d polynomial of sparsity 2^ω(d) can 1/3-approximate a Boolean function. We prove this conjecture unconditionally for a particular class of polynomials.
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