TY - JOUR

T1 - Source Resolvability and Intrinsic Randomness

T2 - Two Random Number Generation Problems with Respect to a Subclass of f-Divergences

AU - Nomura, Ryo

N1 - Funding Information:
Manuscript received August 1, 2019; revised March 21, 2020; accepted July 6, 2020. Date of publication July 14, 2020; date of current version November 20, 2020. This work was supported in part by JSPS KAKENHI under Grant JP18K04150 and in part by the Waseda University Grant for Special Research Projects under Project 2020C-528. This article was presented in part at the 2019 IEEE International Symposium on Information Theory.

PY - 2020/12

Y1 - 2020/12

N2 - This paper deals with two typical random number generation problems in information theory. One is the source resolvability problem (resolvability problem for short) and the other is the intrinsic randomness problem. In the literature, optimum achievable rates in these two problems with respect to the variational distance as well as the Kullback-Leibler (KL) divergence have already been analyzed. On the other hand, in this study we consider these two problems with respect to f-divergences. The f-divergence is a general non-negative measure between two probabilistic distributions on the basis of a convex function f. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to f-divergences. In this paper, we impose some conditions on the function f so as to simplify the analysis, that is, we consider a subclass of f-divergences. Then, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences. Next, we particularize our general formulas to several specified functions f. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. The second-order optimum achievable rates and optimistic optimum achievable rates have also been investigated.

AB - This paper deals with two typical random number generation problems in information theory. One is the source resolvability problem (resolvability problem for short) and the other is the intrinsic randomness problem. In the literature, optimum achievable rates in these two problems with respect to the variational distance as well as the Kullback-Leibler (KL) divergence have already been analyzed. On the other hand, in this study we consider these two problems with respect to f-divergences. The f-divergence is a general non-negative measure between two probabilistic distributions on the basis of a convex function f. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to f-divergences. In this paper, we impose some conditions on the function f so as to simplify the analysis, that is, we consider a subclass of f-divergences. Then, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences. Next, we particularize our general formulas to several specified functions f. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. The second-order optimum achievable rates and optimistic optimum achievable rates have also been investigated.

KW - f-divergence

KW - general source

KW - information-spectrum methods

KW - intrinsic randomness

KW - Kullback-Leibler divergence

KW - source resolvability

KW - variational distance

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U2 - 10.1109/TIT.2020.3009208

DO - 10.1109/TIT.2020.3009208

M3 - Article

AN - SCOPUS:85097348521

VL - 66

SP - 7588

EP - 7601

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 12

M1 - 9140025

ER -