On Smooth R?nyi Entropies: A Novel Information Measure, One-Shot Coding Theorems, and Asymptotic Expansions
On Smooth R?nyi Entropies: A Novel Information Measure, One-Shot Coding Theorems, and Asymptotic Expansions
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关于平滑 R?nyi 熵：一种新颖的信息测度、一次性编码定理和渐近展开式
DOI：
10.1109/tit.2021.3132670
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发表时间：
2022
影响因子：
2.5
通讯作者：
Tan Vincent Y. F.
中科院分区：
文献类型：
--
作者：
Sakai Yuta;Tan Vincent Y. F.
本研究考虑了Renner和Wolf [ASIACRYPT， 2005]提出的无条件光滑r<s:1>尼伊熵，Kuzuoka [IEEE Trans. cn]提出的光滑条件r<s:1>尼伊熵。正,Th。， 66(3), 1674-1690, 2020]，以及一个我们称之为条件光滑的新量——百科熵。在没有侧信息的情况下，后两个量可以专门化为第一个量。我们通过建立Campbell的源编码问题、Arıkan-Massey猜测问题和Bunte-Lapidoth任务编码问题等信息论问题的一次性编码定理，探讨了这些光滑r<s:1>熵的操作作用。我们在误差不消失的情况下考虑这些问题，对于每个问题，我们考虑两种误差形式：平均和最大误差标准，其中平均和最大化是关于侧信息的。利用单次编码定理，我们得出Kuzuoka光滑条件r<s:1>尼伊熵和条件光滑- -熵分别是涉及平均和最大误差准则的问题的解。此外，我们研究了这些熵的渐近展开式，当潜在的源及其侧信息是平稳的和无记忆的。应用单次编码定理的渐近展开式，我们得到了这些问题的各种基本极限。我们证明，在非简并设置下，一阶基本极限在平均和最大误差准则下是不同的。这与当前作者所考虑的不同但相关的设置形成对比[IEEE Trans]。正,Th。[j], 66(12), 7565-7587, 2020]，用于允许错误的变长条件源编码，在这些错误准则下，一阶项相同但二阶项不同。
This study considers the unconditional smooth Rényi entropy proposed by Renner and Wolf [ASIACRYPT, 2005], the smooth conditional Rényi entropy proposed by Kuzuoka [IEEE Trans. Inf. Th., 66(3), 1674–1690, 2020], and a novel quantity which we term theconditional smooth-⋆entropy.The latter two quantities can be specialized to the first in the absence of side-information. We explore the operational roles of these smooth Rényi entropies by establishing one-shot coding theorems for several information-theoretic problems, including Campbell’s source coding problem, the Arıkan–Massey guessing problem, and the Bunte–Lapidoth task encoding problem. We consider these problems in cases where the errors are non-vanishing and for each problem, we consider two error formalisms: the average and maximum error criteria, where the averaging and maximization are taken with respect to the side-information. Using the one-shot coding theorems, we conclude that Kuzuoka’s smooth conditional Rényi entropy and the conditional smooth-⋆ entropy are the solutions to the problems involving the average and maximum error criteria, respectively. Furthermore, we examine asymptotic expansions of these entropies when the underlying source with its side-information is stationary and memoryless. Applying our asymptotic expansions to the one-shot coding theorems, we derive various fundamental limits for these problems. We show that, under non-degenerate settings, the first-order fundamental limits differ under the average and maximum error criteria. This is in contrast to a different but related setting considered by the present authors [IEEE Trans. Inf. Th., 66(12), 7565–7587, 2020], for variable-length conditional source coding allowing errors, in which the first-order terms are identical but the second-order terms are different under these error criteria.