On the $Φ$-Stability and Related Conjectures

Abstract

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 ρ f (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.

Common Information, Noise Stability, and Their Extensions

“…Using Fourier analysis and optimization theory, the first author of this monograph [191] provided an explicit threshold for Theorem 9.4.1. Specifically, he showed that (9.45) holds for any n and any ρ ∈ (0, ρ 1 ], where ρ 1 be the solution in (0, 1) to the equation…”

“…Since the Li-Médard conjecture was only recently posed (at the time of writing), there is less progress on it compared to the Courtade-Kumar conjecture. Hence, we do not elaborate on it apart from mentioning some partial progress by Yu [191] for a certain set of (q, ρ).…”

“…There is even less progress for the asymmetric case in which the best known result remains that of Witsenhausen's result in Corollary 9.2.1 for the case q = 2. Recently, in [191], the first author of this monograph made some progress on the Mossel-O'Donnell conjecture. He showed that the symmetric version of the Mossel-O'Donnell conjecture holds for 2 < q ≤ 5, and the asymmetric version holds for 2 < q ≤ 3.…”