The "Most informative boolean function" conjecture holds for high noise

Alex Samorodnitsky

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arXiv · 1510.08656

The "Most informative boolean function" conjecture holds for high noise

Alex Samorodnitsky

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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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Alex Samorodnitsky. 2015-10-29. The "Most informative boolean function" conjecture holds for high noise. https://arxiv.org/abs/1510.08656

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