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- W1789240725 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." @default.
- W1789240725 created "2016-06-24" @default.
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- W1789240725 date "2015-06-10" @default.
- W1789240725 modified "2023-09-27" @default.
- W1789240725 title "Remarks on the Most Informative Function Conjecture at fixed mean" @default.
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