AI 中文总结
研究输入字母表为[0,1]、输出字母表为{0,…,n}的二项式信道，通过建立容量实现输入分布的结构性质等方法，得出容量上下界及支持大小下界，证明贝塔 - 二项式输出渐近最优等。
AI 中文摘要
我们研究输入字母表为[0,1]且输出字母表为{0,…,n}的二项式信道。研究其容量及容量实现输入和输出分布的结构。因输出字母表有限而输入字母表连续，不同输入分布可能诱导相同输出分布。我们首先建立容量实现输入分布的结构性质，推导容量的显式非渐近上下界，证明关于容量实现输入分布支持大小的改进下界等。数值结果说明了容量界和最优输入分布。
英文摘要
We study the binomial channel with input alphabet $[0,1]$ and output alphabet ${0,\ldots,n}$. We investigate its capacity and the structure of the capacity-achieving input and output distributions. Since the output alphabet is finite whereas the input alphabet is continuous, different input distributions may induce the same output distribution; hence, uniqueness and support properties of optimal inputs do not follow from strict concavity arguments. We first establish structural properties of the capacity-achieving input distribution. In particular, we show that it is discrete, unique, symmetric around $1/2$, and contains the endpoints ${0,1}$ in its support. We also derive location constraints and bounds on the probability masses of support points, and improve the Witsenhausen-type upper bound on the support size from order $n$ to order $n/2$. We derive explicit nonasymptotic upper and lower bounds on the capacity $C(n)$. These bounds imply $C(n)=\frac{1}{2}\log(\frac{nπ}{2e})+o(1).$ The lower bound is obtained by evaluating the mutual information at the reference input $X_r\sim \mathrm{Beta}(1/2,1/2)$, which induces a beta-binomial output distribution, while the upper bound follows from a minimax redundancy construction. Finally, we prove an improved lower bound on the support size of the capacity-achieving input distribution. We show that the beta-binomial output induced by $X_r$ is asymptotically optimal and close to the capacity-achieving output distribution in relative entropy and $χ^2$ divergence. We also prove a finite-mixture approximation lower bound showing that the beta-binomial output cannot be approximated too accurately by binomial mixtures with few components. Combining these results yields a support-size lower bound of order $Ω(\sqrt{n\log\log n})$, with explicit constants. Numerical results illustrate the capacity bounds and optimal input.
CommentsarXiv admin note: substantial text overlap with arXiv:2401.12818
