基于成对错误概率的通用译码
Universal Decoding via the Pairwise Error Probability
- Tel Aviv University（特拉维夫大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文基于成对错误概率提出通用译码理论，通过截断逆PEP统计量合并多度量谱，实现随机编码通用性，并扩展到擦除与连续信道。
AI中文摘要：
我们发展了一种基于成对错误概率（PEP）的通用译码理论，该PEP是伴随论文中的基本概念。对于任意译码度量，我们定义了PEP及其错误谱，并且虽然一个家族中的原始度量值没有共同的尺度，但PEP提供了一个。通用译码——即同时针对整个度量或信道家族进行良好译码——成为将每个度量的谱合并为单一规则的问题，我们通过一个截断的逆PEP统计量来构建该规则。该构建基于一个适用于任何输入先验的Kraft型译码不等式：对于每个输出，每个度量被规范化为对码字的条件概率分配，而合并是诱导家族的归一化最大似然包络。我们证明了该规则是随机编码通用的（相对于家族中的最佳度量，它仅损失一个可忽略的速率，针对每个信道），表明它是一个渐近最小最大/等化器规则，并对其进行去随机化：在次指数信道家族上，单个确定性码继承该保证。对于离散无记忆信道，该构建简化为类型，并恢复了最大互信息译码器，其倾斜变体用于非均匀无记忆输入，以及有限状态通用译码器。它扩展到具有擦除选项的译码（确定性地，在擦除余量上均匀），并通过可分度量家族的离散化扩展到连续字母表：对于具有确定性干扰的AWGN，它达到匹配ML指数，直到一个显式的典型集上限，对于ISI信道，在等化器和信道谱的显式假设以及相同的上限下，它达到等化-译码规则家族中的最佳指数。在整个过程中，通用性是PEP分析的推论，而不是一个单独的理论。
英文摘要：
We develop a theory of universal decoding built on the pairwise error probability (PEP) primitive of a companion paper. The PEP and its error spectrum are defined for an arbitrary decoding metric, and while raw metric values across a family share no common scale, the PEP supplies one. Universal decoding -- decoding well simultaneously against a whole family of metrics or channels -- becomes the problem of merging the per-metric spectra into a single rule, which we construct from a clipped inverse-PEP statistic. The construction rests on a Kraft-type inequality for decoding, valid for any input prior: per output, every metric canonicalizes into a conditional probability assignment on the codewords, and the merge is the normalized-maximum-likelihood envelope of the induced family. We prove the rule is random-coding universal (it loses only a vanishing rate relative to the best metric in the family, against every channel), show it is an asymptotic minimax/equalizer rule, and derandomize it: over a subexponential channel family a single deterministic code inherits the guarantee. For discrete memoryless channels the construction reduces to types and recovers the maximum-mutual-information decoder, its tilted variant for non-uniform memoryless input, and finite-state universal decoders. It extends to decoding with an erasure option (deterministically, uniformly over erasure margins) and, via a discretization of separable metric families, to continuous alphabets: for AWGN with deterministic interference it attains the matched-ML exponent up to an explicit typical-set cap, and for ISI channels the best exponent in a family of equalize-and-decode rules, under explicit assumptions on the equalizers and the channel spectrum and the same cap. Throughout, universality is a corollary of the PEP analysis rather than a separate theory.
