Optimum Achievable Rates in Two Random Number Generation Problems with $f$-Divergences Using Smooth Rényi Entropy

TL;DR

The paper addresses optimum achievable rates for two fixed-length random number generation problems—the source resolvability and intrinsic randomness—under a broad class of $f$-divergences. It develops a unified approach using smooth Rényi entropy to derive first- and second-order rate formulas, and introduces an offset function $f_0$ to broaden the $f$-divergence class considered. The results reveal explicit rate expressions for many important distances (e.g., KL, variational, Hellinger, $E__gamma$) and establish a duality between the two problems in this framework. The findings offer practical pathways to compute achievable rates across diverse measures and link information-spectrum quantities with smooth Rényi entropies, enriching the theoretical understanding of random-number generation under approximation constraints.

Abstract

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 of two 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.

Figures (4)

• Figure 1: Smooth max entropy $H_0(\delta|X^n)$
• Figure 2: Smooth min entropy $H_\infty(\delta|X^n)$
• Figure 3: Resolvability problem
• Figure 4: Intrinsic randomness problem

Theorems & Definitions (62)

• Definition 2.1: $f$-divergence csiszar2004information
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.2: Smooth Rényi entropy of order $\alpha$ RW2004
• Theorem 2.1: Uyematsu Uyematsu_ISIT2010Uyematsu2010
• Theorem 2.2: Uyematsu and Kunimatsu Uyematsu13
• Definition 3.1