Emergent Mind
The "Most informative boolean function" conjecture holds for high noise
(1510.08656)
Published Oct 29, 2015
in
cs.IT
,
math.CO
,
math.IT
,
and
math.PR
Abstract
We prove the "Most informative boolean function" conjecture of Courtade and Kumar for high noise $\epsilon \ge 1/2 - \delta$, for some absolute constant $\delta > 0$. Namely, if $X$ is uniformly distributed in ${0,1}n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $\epsilon$, then, provided $\epsilon \ge 1/2 - \delta$, for any boolean function $f$ holds $I(f(X);Y) \le 1 - H(\epsilon)$. This conjecture was previously known to hold only for balanced functions.
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