Remarks on the Most Informative Function Conjecture at fixed mean

Guy Kindler; Ryan O'Donnell; David Witmer

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arXiv · 1506.03167

Remarks on the Most Informative Function Conjecture at fixed mean

Guy Kindler·Ryan O'Donnell·David Witmer

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Abstract

In 2013, Courtade and Kumar posed the following problem: Let $\boldsymbol{x} \sim \{\pm 1\}^n$ be uniformly random, and form $\boldsymbol{y} \sim \{\pm 1\}^n$ by negating each bit of $\boldsymbol{x}$ independently with probability $\alpha$. Is it true that the mutual information $I(f(\boldsymbol{x}) \mathbin{;} \boldsymbol{y})$ is maximized among $f:\{\pm 1\}^n \to \{\pm 1\}$ by $f(x) = x_1$? We do not resolve this problem. Instead, we make a couple of observations about the fixed-mean version of the conjecture. We show that Courtade and Kumar's stronger Lex Conjecture fails for small noise rates. We also prove a continuous version of the conjecture on the sphere and show that it implies the previously-known analogue for Gaussian space.

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Guy Kindler, Ryan O'Donnell, David Witmer. 2015-06-10. Remarks on the Most Informative Function Conjecture at fixed mean. https://arxiv.org/abs/1506.03167

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