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.
Publisher Copyright:
© 1963-2012 IEEE.
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 - Kullback-Leibler divergence
KW - f-divergence
KW - general source
KW - information-spectrum methods
KW - intrinsic randomness
KW - source resolvability
KW - variational distance
UR - http://www.scopus.com/inward/record.url?scp=85097348521&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85097348521&partnerID=8YFLogxK
U2 - 10.1109/TIT.2020.3009208
DO - 10.1109/TIT.2020.3009208
M3 - Article
AN - SCOPUS:85097348521
SN - 0018-9448
VL - 66
SP - 7588
EP - 7601
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 12
M1 - 9140025
ER -