用于单次信息论的成对错误概率框架
A Pairwise-Error-Probability Framework for One-Shot Information Theory
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中文总结 AI 辅助
该研究提出基于成对错误概率的单次信息论信道编码框架，推导相关变分恒等式，可恢复经典单次界，通过数值示例验证其效果。
中文摘要 AI 辅助
我们基于采用随机平局规则的译码器的成对错误概率（PEP），构建了一个单次（有限码长）信道编码框架。该平局规则可得到一个概率积分变换恒等式：对于任意译码度量，由此产生的错误频谱既描述了随机编码可达性，又给出了精确的固定码逆命题。我们针对PEP的度量加权尾泛函推导了两个变分恒等式，分别通过奈曼-皮尔逊β泛函和反向信道得到，适用于任意度量。在匹配的最大似然译码下，这些恒等式会特化为频谱本身的表示，而该频谱在检验水平和输入先验上是联合凸的；结合反向信道表示，这为经先验优化的极小极大元逆命题提供了一个线性规划，该规划一般为有限维，且对于具有固定字母的无记忆信道，在类型归约后其规模为码长的多项式。随机编码界的先验优化被表述为输入分布上的凹规划，具有显式梯度，可通过直接一阶方法求解。该框架可恢复多个经典单次界，包括Polyanskiy-Poor-Verdu的随机编码联合界和极小极大元逆命题、Han-Verdu的信息谱界以及Matthews的线性规划逆命题。对加性高斯白噪声（AWGN）和二进制Z信道的数值示例说明了可达性-逆命题比较以及先验优化的效果。
英文摘要
We develop a one-shot (finite-blocklength) channel-coding framework based on the pairwise error probability (PEP) of a decoder with randomized tie-breaking. The tie-breaking rule yields a probability-integral-transform identity: the induced error spectrum describes both random-coding achievability and exact fixed-code converse statements, for an arbitrary decoding metric. We derive two variational identities for metric-weighted tail functionals of the PEP, one through the Neyman-Pearson $β$-functional and one through a reverse channel, valid for an arbitrary metric. Under matched maximum-likelihood decoding they specialize to representations of the spectrum itself, which is then jointly convex in the testing level and the input prior; combined with the reverse-channel representation, this gives a linear program for the prior-optimized minimax meta-converse -- finite-dimensional in general and, for memoryless channels with fixed alphabets, of size polynomial in the blocklength after a type reduction. Prior optimization of the random-coding bound is formulated as a concave program over input distributions with an explicit gradient, solved by a direct first-order method. The framework recovers several classical one-shot bounds, including the random-coding union bound and minimax meta-converse of Polyanskiy-Poor-Verdu, the information-spectrum bounds of Han-Verdu, and the linear-programming converse of Matthews. Numerical examples on the AWGN and binary Z-channels illustrate the achievability-converse comparison and the effect of prior optimization.
