On the Φ-Stability and Related Conjectures
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Let 𝐗 be a random variable uniformly distributed on the discrete cube { -1,1} ^n, and let T_ρ be the noise operator acting on Boolean functions f:{ -1,1} ^n→{ 0,1}, where ρ∈[0,1] is the noise parameter, representing the correlation coefficient between each coordination of 𝐗 and its noise-corrupted version. Given a convex function Φ and the mean 𝔼f(𝐗)=a∈[0,1], which Boolean function f maximizes the Φ-stability 𝔼[Φ(T_ρf(𝐗))] of f? Special cases of this problem include the (symmetric and asymmetric) α-stability problems and the "Most Informative Boolean Function" problem. In this paper, we provide several upper bounds for the maximal Φ-stability. Considering specific Φ's, we partially resolve Mossel and O'Donnell's conjecture on α-stability with α>2, Li and Médard's conjecture on α-stability with 1<α<2, and Courtade and Kumar's conjecture on the "Most Informative Boolean Function" which corresponds to a conjecture on α-stability with α=1. Our proofs are based on discrete Fourier analysis, optimization theory, and improvements of the Friedgut–Kalai–Naor (FKN) theorem. Our improvements of the FKN Theorem are sharp or asymptotically sharp for certain cases.
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