Emergent Mind
An Improved Upper Bound for the Most Informative Boolean Function Conjecture
(1505.05794)
Published May 21, 2015
in
cs.IT
and
math.IT
Abstract
Suppose $X$ is a uniformly distributed $n$-dimensional binary vector and $Y$ is obtained by passing $X$ through a binary symmetric channel with crossover probability $\alpha$. A recent conjecture by Courtade and Kumar postulates that $I(f(X);Y)\leq 1-h(\alpha)$ for any Boolean function $f$. So far, the best known upper bound was $I(f(X);Y)\leq (1-2\alpha)2$. In this paper, we derive a new upper bound that holds for all balanced functions, and improves upon the best known bound for all $\tfrac{1}{3}<\alpha<\tfrac{1}{2}$.
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