On Smooth Rényi Entropies: A Novel Information Measure, One-Shot Coding Theorems, and Asymptotic Expansions

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 the conditional smooth -⋆ entropy. The latter two qua...

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Vydáno v:IEEE transactions on information theory Ročník 68; číslo 3; s. 1496 - 1531
Hlavní autoři: Sakai, Yuta, Tan, Vincent Y. F.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.03.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Arıkan–Massey guessing problems
Asymptotic methods
Asymptotic series
Bunte–Lapidoth encoding tasks
Channel coding
Codes
Coding
Criteria
cumulant generating function of codeword lengths
Entropy
Information theory
Proposals
second-order asymptotics
Smooth Rényi entropy
Source coding
Task analysis
Theorems
Upper bound
ISSN:0018-9448, 1557-9654
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Shrnutí: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 the conditional 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.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2021.3132670