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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Vydané v:	IEEE transactions on information theory Ročník 68; číslo 3; s. 1496 - 1531
Hlavní autori:	Sakai, Yuta, Tan, Vincent Y. F.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York IEEE 01.03.2022 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:	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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Abstract	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.
AbstractList	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.
Author	Sakai, Yuta Tan, Vincent Y. F.
Author_xml	– sequence: 1 givenname: Yuta orcidid: 0000-0003-0183-6988 surname: Sakai fullname: Sakai, Yuta email: yuta.sakai@eng.u-hyogo.ac.jp organization: Department of Electronics and Computer Science, Graduate School of Engineering, University of Hyogo, Himeji, Japan – sequence: 2 givenname: Vincent Y. F. orcidid: 0000-0002-5008-4527 surname: Tan fullname: Tan, Vincent Y. F. email: vtan@nus.edu.sg organization: Department of Electrical and Computer Engineering and Department of Mathematics, National University of Singapore, Singapore
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SubjectTerms	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
Title	On Smooth Rényi Entropies: A Novel Information Measure, One-Shot Coding Theorems, and Asymptotic Expansions
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