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Asia Pacific Workshop on Data Science and Information Theory (APWDSIT)

A Note on Channel Resolvability Rate with Subclass of $f$-Divergences

Ryo Nomura, Hideki Yagi

Published Oct 20, 2025• 0 citations
Publisher

Abstract

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 of$f$-divergences. The$f$-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 to$f$-divergence provides valuable insights. We derive both upper and lower bounds for the optimal achievable rate for a specific subclass of$f$-divergence. These bounds enable us to determine the exact$\varepsilon$-optimal rate for the variational distance, while also establishing the exact 0 -optimal rate for various$f$-divergences such as the Hellinger distance, JensenShannon divergence, and other measures within this framework.

DOI
10.1109/apwdsit64675.2025.11478540

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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.
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 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.
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.
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.
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.
The conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy. In contrast to previous works, in this paper, we use the (normalized or unnormalized) Rényi divergence (with the Rényi parameter in [0,2]∪{∞}[0,2]\cup\{\infty\}[0,2]∪{∞}) to measure the level of approximation. We also provide asymptotic expressions for normalized Rényi divergence when the Rényi parameter is larger than or equal to 111 as well as (lower and upper) bounds for the case when the same parameter is smaller than 111. We characterize the Rényi resolvability, which is defined as the minimum rate required to ensure that the Rényi divergence vanishes asymptotically. The Rényi resolvabilities are the same for both the normalized and unnormalized divergence cases. In addition, when the Rényi parameter smaller than~111, consistent with the traditional case where the Rényi parameter is equal to~111, the Rényi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution. When the Rényi parameter is larger than 111 the Rényi resolvability is, in general, larger than the mutual information. The optimal Rényi divergence is proven to vanish at least exponentially fast for both of these two cases, as long as the code rate is larger than the Rényi resolvability. The optimal exponential rate of decay for i.i.d.\ random codes is also characterized exactly. We apply these results to the wiretap channel, and completely characterize the optimal tradeoff between the rates of the secret and non-secret messages when the leakage measure is given by the (unnormalized) Rényi divergence.