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arXiv · 1807.11289

On the Most Informative Boolean Functions of the Very Noisy Channel

Hengjie Yang·Richard D. Wesel
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Abstract

Let $X^n$ be a uniformly distributed $n$-dimensional binary vector, and $Y^n$ be the result of passing $X^n$ through a binary symmetric channel (BSC) with crossover probability $\alpha$. A recent conjecture postulated by Courtade and Kumar states that for any Boolean function $f:\{0,1\}^n\to\{0,1\}$, $I(f(X^n);Y^n)\le 1-H(\alpha)$. Although the conjecture has been proved to be true in the dimension-free high noise regime by Samorodnitsky, here we present a calculus-based approach to show a dimension-dependent result by examining the second derivative of $H(\alpha)-H(f(X^n)|Y^n)$ at $\alpha=1/2$. Along the way, we show that the dictator function is the most informative function in the high noise regime.

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Hengjie Yang, Richard D. Wesel. 2018-07-30. On the Most Informative Boolean Functions of the Very Noisy Channel. https://arxiv.org/abs/1807.11289

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