The maximum mutual information between the output of a discrete symmetric channel and several classes of Boolean functions of its input
Septimia Sarbu
备注：
Withdrawn due to rejection by the IEEE Transactions on Information Theory; the reason for rejection was limited originality and scope of the proposed result
我们证明了Courtade-Kumar猜想，几类n维布尔函数，所有$n \geq 2$和所有值的错误概率的二进制对称信道，$0 \leq p \leq 1/2$。该猜想指出，无记忆二进制对称通道的独立且同分布输入的n维向量的任何布尔函数与相应的输出向量之间的互信息的上限为$1-\operatorname{H}(p)$，其中$\operatorname{H}(p)$表示二进制熵函数。也就是说，令$\mathbf{X}=[X_1 \ldots X_n]$是独立且同分布的伯努利（1/2）随机变量的向量，其是无记忆二进制对称信道的输入，其中误差概率在$0 \leq p \leq 1/2$和$\mathbf{Y}=[Y_1 \ldots Y_n]$的区间中，是对应的输出。设$f:\{0,1\}^n \rightarrow \{0,1\}$为n维布尔函数。然后是$\operatorname{MI}(f(X),Y) \leq 1-\operatorname{H}(p)$。我们的证明采用Karamata定理，概率论的概念，随机变量和向量的变换和代数运算。
英文摘要：
We prove the Courtade-Kumar conjecture, for several classes of n-dimensional Boolean functions, for all $n \geq 2$ and for all values of the error probability of the binary symmetric channel, $0 \leq p \leq 1/2$. This conjecture states that the mutual information between any Boolean function of an n-dimensional vector of independent and identically distributed inputs to a memoryless binary symmetric channel and the corresponding vector of outputs is upper-bounded by $1-\operatorname{H}(p)$, where $\operatorname{H}(p)$ represents the binary entropy function. That is, let $\mathbf{X}=[X_1 \ldots X_n]$ be a vector of independent and identically distributed Bernoulli(1/2) random variables, which are the input to a memoryless binary symmetric channel, with the error probability in the interval $0 \leq p \leq 1/2$ and $\mathbf{Y}=[Y_1 \ldots Y_n]$ the corresponding output. Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be an n-dimensional Boolean function. Then, $\operatorname{MI}(f(X),Y) \leq 1-\operatorname{H}(p)$. Our proof employs Karamata's theorem, concepts from probability theory, transformations of random variables and vectors and algebraic manipulations.
Optimal Computation of Symmetric Boolean Functions in Collocated Networks
We consider collocated wireless sensor networks, where each node has a Boolean measurement and the goal is to compute a given Boolean function of these measurements. We first consider the worst case setting and study optimal block computation strategies for computing symmetric Boolean functions. We study three classes of functions: threshold functions, delta functions and interval functions. We provide exactly optimal strategies for the first two classes, and a scaling law order-optimal strategy with optimal preconstant for interval functions. We also extend the results to the case of integer …
PAC-Bayes, MAC-Bayes and Conditional Mutual Information: Fast rate bounds that handle general VC classes
CWI Amsterdam（阿姆斯特丹数学与计算机科学国家研究所）；Leiden University（莱顿大学）；Google Research（谷歌研究院）；Khoury College of Computer Sciences, Northeastern University（东北大学库里计算机科学学院）
