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IEEE Information Theory Workshop (ITW)

Intrinsic Randomness Problem with Respect to a Subclass of f-divergence

Ryo Nomura

Published Aug 1, 2019• 2 citations
Publisher

Abstract

This paper deals with the intrinsic randomness (IR) problem, which is one of typical random number generation problems. In the literature, the optimum achievable rates in the IR problem 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 the IR problem with respect to a subclass of f-divergences. The f-divergence is a general non-negative measure between two probabilistic distributions and includes several important measures such as the total variational distance, the X2-divergence, the KL divergence, and so on. Hence, it is meaningful to consider the IR problem with respect to the f-divergence. In this paper, we assume some conditions on the f-divergence for simplifying the analysis. That is, we focus on a subclass of f-divergences. In this problem setting, we first derive the general formula of the optimum achievable rate. Next, we show that it is easy to derive the optimum achievable rate with respect to the variational distance, the KL divergence, and the Hellinger distance from our general formula.

DOI
10.1109/itw44776.2019.8989245

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This paper deals with the intrinsic randomness (IR) problem, which is one of typical random number generation problems. In the literature, the optimum achievable rates in the IR problem 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 the IR problem with respect to a subclass of f-divergences. The f-divergence is a general non-negative measure between two probabilistic distributions and includes several important measures such as the total variational distance, the X2-divergence, the KL divergence, and so on. Hence, it is meaningful to consider the IR problem with respect to the f-divergence. In this paper, we assume some conditions on the f-divergence for simplifying the analysis. That is, we focus on a subclass of f-divergences. In this problem setting, we first derive the general formula of the optimum achievable rate. Next, we show that it is easy to derive the optimum achievable rate with respect to the variational distance, the KL divergence, and the Hellinger distance from our general formula.
This paper deals with the source resolvability problem which is one of typical random number generation problems. In the literatures, the achievable rate in the source resolvability problem with respect to the variational distance as well as the Kullback-Leibler (KL) divergence have been analyzed. On the other hand, in this study we consider the problem with respect to a subclass of f-divergence. The f-divergence is a general non-negative measure between two probabilistic distributions and 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 source resolvability problem with respect to the f-divergence. We derive the general formula of the optimum achievable rate for a certain subclass of the f-divergence. Then, we reveal that it is easy to derive previous results from our general formula.
The intrinsic randomness (IR) problem is considered for general setting. In the literature, the optimum IR rate with respect to the variational distance has been characterized in two ways. One is based on the information spectrum quantity and the other is based on the smooth Rényi entropy. Recently, Nomura has revealed the optimum IR rate with respect tofff-divergences, which includes the variational distance, the Kullback-Leibler (KL) divergence and so on, by using the informational spectrum quantity. In this paper, we try to characterize the optimum IR rate with respect to a subclass offff-divergences by using the smooth Min entropy. The subclass offff-divergences considered in this paper includes typical distance measures such as the total variational distance, the KL divergence, the Hellinger distance and so on.
Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth Rényi entropy. Recently, optimum achievable rates with respect to f-divergences have been characterized using the information spectrum quantity. The f-divergence is a general non-negative measure between two probability 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. However, optimum achievable rates with respect to f-divergences using the smooth Rényi entropy have not been clarified yet in both problems. In this paper, we try to analyze the optimum achievable rates using the smooth Rényi entropy and to extend the class of f-divergence. To do so, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on f-divergence and generalize the obtained general formulas. Then, 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. Furthermore, a kind of duality between the resolvability and the intrinsic randomness is revealed in terms of the smooth Rényi entropy. Second-order optimum achievable rates and optimistic achievable rates are also investigated.
The source resolvability problem (or resolvability problem for short) is one of random number generation problems in information theory. In the literature, the optimum achievable rates in the resolvability problem have been characterized in different two ways. One is based on the information spectrum quantity and the other is based on the smooth Rényi entropy. Recently, Nomura has revealed the optimum achievable rate with respect to the f-divergence, which includes the variational distance, the Kullback-Leibler (KL) divergence and so on. On the other hand, the optimum achievable rates with respect to the variational distance has been characterized by using the smooth Rényi entropy. In this paper, we try to extend this result to the case of other distances. To do so, we consider the resolvability problem with respect to the subclass of f-divergences and determine the optimum achievable rate in terms of the smooth Rényi entropy. The subclass of f-divergences considered in this paper includes typical distance measures such as the total variational distance, the KL divergence, the Hellinger distance and so on.
A Note on Channel Resolvability Rate with Subclass of fff-Divergences
This paper investigates the channel resolvability problem in a general setting. Previous studies have primarily focused on the channel resolvability problem with respect to the variational distance. In this study, we extend this analysis to a subclass offff-divergences. Thefff-divergence is a general non-negative measure between two probability distributions, encompassing several important measures including the variational distance, KL divergence, and Hellinger distance. Given this generality, analyzing the channel resolvability problem with respect tofff-divergence provides valuable insights. We derive both upper and lower bounds for the optimal achievable rate for a specific subclass offff-divergence. These bounds enable us to determine the exactε\varepsilonε-optimal rate for the variational distance, while also establishing the exact 0 -optimal rate for variousfff-divergences such as the Hellinger distance, JensenShannon divergence, and other measures within this framework.
The first- and second-order optimum achievable rates in the source resolvability problem are considered for general sources. In the literature, the achievable rates in the resolvability problem with respect to the variational distance as well as the normalized Kullback-Leibler (KL) divergence have already been analyzed. On the other hand, in this study we consider the source resolvability problem with respect to (unnormalized) KL divergence and derive general formulas of the first- and second-order optimum achievable rates. Relationships with other problems in information theory have also been discussed.