信道不确定性下的软GRAND：极小极大后验包络排序与Gallager指数保持

中文摘要 AI 辅助

软猜测加性噪声解码以依赖于观测的顺序查询校正。对于已知信道，匹配顺序按条件后验概率的非递增顺序对校正进行排序。在信道不确定性下，可容许信道可能诱导匹配顺序，使得同一观测下同一校正被赋予不同秩次。我们转而采用后验包络顺序，该顺序由码本无关的均匀输入校正后验的逐点上确界定义。对于固定的可测后验代表和可测包络，该顺序在每个观测处最小化具有正后验概率的可容许信道-校正对上秩-后验乘积的最坏情况对数。对于均匀子集码本，这一有限块极小极大性质本身并不能确定系综平均误差指数，该指数取决于实现校正秩的分布。对于具有有限校正字母表的独立同分布校正-观测对，其联合分布由该字母表上的计数测度与观测空间上的σ-有限测度的乘积控制，Arimoto条件Rényi熵刻画了匹配秩矩谱。当辅助后验包络顺序的后验族包含相应的条件幂倾斜，并且在观测集上具有亚指数条件Shtarkov复杂度（其补概率以比校正空间基数的倒数更快的指数速度衰减）时，该顺序在[0,1]上保持此谱。在额外的可微性和熵非退化条件下，辅助顺序在归一化码率严格介于零和一之间的每个固定值处达到匹配的均匀子集系综平均误差指数。对于均匀输入无记忆信道，该指数等于Gallager的均匀输入随机编码指数。

英文摘要

Soft Guessing Random Additive Noise Decoding queries corrections in an observation-dependent order. For a known channel, the matched order ranks corrections by nonincreasing conditional posterior probability. Under channel uncertainty, admissible channels may induce matched orders that assign different ranks to the same correction at the same observation. We instead use a posterior-envelope order, defined by the pointwise supremum of codebook-independent uniform-input correction posteriors. For fixed measurable posterior representatives and a measurable envelope, this order minimizes, at each observation, the worst-case logarithm of the rank--posterior product over admissible channel--correction pairs with positive posterior probability. For uniform-subset codebooks, this finite-block minimax property does not by itself determine the ensemble-average error exponent, which depends on the distribution of the realized correction rank. For i.i.d. correction--observation pairs with a finite correction alphabet whose joint distribution is dominated by the product of counting measure on that alphabet and a $σ$-finite measure on the observation space, Arimoto conditional Rényi entropy characterizes the matched rank-moment spectrum. An auxiliary posterior-envelope order preserves this spectrum on $[0,1]$ when its posterior family contains the corresponding conditional power tilts and has subexponential conditional Shtarkov complexity on observation sets whose complement probabilities decay exponentially faster than the reciprocal correction-space cardinality. Under additional differentiability and entropy nondegeneracy conditions, the auxiliary order attains the matched uniform-subset ensemble-average error exponent at every fixed normalized code rate strictly between zero and one. For uniform-input memoryless channels, this exponent equals Gallager's uniform-input random-coding exponent.

发表机构

• Harvard University（哈佛大学）

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