关于噪声函数的熵
On the entropy of a noisy function
摘要：
令$0 < ε< 1/2$为噪声参数，并令$T_ε$为作用于布尔立方体$\{0,1\}^n$上的函数的噪声算子。设$f$是$\{0,1\}^n$上的非负函数。我们上界的熵$T_ε f$的平均熵的条件期望的$f$，给定的集合大致$(1-2ε)^2 \cdot n$变量。在信息理论方面，我们证明了以下加强“格伯夫人引理”：让$X$是一个随机二进制向量的长度$n$，并让$Z$是一个噪声向量，对应于一个二进制对称信道的交叉概率$ε$。然后，设置$v = (1-2ε)^2 \cdot n$，我们有（到低阶项）：$$ H\Big(X \oplus Z\Big) \ge n \cdot H\left(ε~+~ (1-2ε) \cdot H^{-1}\left(\frac{{\mathbb E}_{|B| = v} H\Big(\{X_i\}_{i\in B}\Big)}{v}\right)\right) $$作为一个应用程序，我们表明，对于一个布尔函数$f$，这是接近的特征功能$g$的一个子立方体的尺寸$n-1$，熵的$T_ε f$是最多的$T_ε g$。结合Ordentlich、Shayevitz和Weinstein最近的结果，Courtade和Kumar的“最具信息量的布尔函数”猜想对于高噪声$ε\ge 1/2 - δ$和某些绝对常数$δ> 0$成立。也就是说，如果$X$均匀分布在$\{0,1\}^n$中，并且$Y$是通过以概率$ε$独立地翻转$X$的每个坐标而获得的，那么，如果$ε\ge 1/2 - δ$对于任何布尔函数$f$保持$I\Big(f(X);Y\Big) \le 1 - H(ε)$。
英文摘要：
Let $0 < ε< 1/2$ be a noise parameter, and let $T_ε$ be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$. We upper bound the entropy of $T_ε f$ by the average entropy of conditional expectations of $f$, given sets of roughly $(1-2ε)^2 \cdot n$ variables. In information-theoretic terms, we prove the following strengthening of "Mrs. Gerber's lemma": Let $X$ be a random binary vector of length $n$, and let $Z$ be a noise vector, corresponding to a binary symmetric channel with crossover probability $ε$. Then, setting $v = (1-2ε)^2 \cdot n$, we have (up to lower-order terms): $$ H\Big(X \oplus Z\Big) \ge n \cdot H\left(ε~+~ (1-2ε) \cdot H^{-1}\left(\frac{{\mathbb E}_{|B| = v} H\Big(\{X_i\}_{i\in B}\Big)}{v}\right)\right) $$ As an application, we show that for a boolean function $f$, which is close to a characteristic function $g$ of a subcube of dimension $n-1$, the entropy of $T_ε f$ is at most that of $T_ε g$. This, combined with a recent result of Ordentlich, Shayevitz, and Weinstein shows that the "Most informative boolean function" conjecture of Courtade and Kumar holds for high noise $ε\ge 1/2 - δ$, for some absolute constant $δ> 0$. Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $ε$, then, provided $ε\ge 1/2 - δ$, for any boolean function $f$ holds $I\Big(f(X);Y\Big) \le 1 - H(ε)$.
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