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Optimum Self Random Number Generation Rate and Its Application to Rate Distortion Perception Function

Ryo Nomura

TL;DR

This work extends SRNG from the variational distance to a broad class of $f$-divergences, providing a general rate formula $S_f(\Delta|{\bf X}) = K_f(\Delta|{\bf X})$ for $0 \le \Delta < f(0)$ and deriving explicit expressions for key divergences (e.g., variational distance, reverse KL, Hellinger, and $E_\gamma$-divergence). It reveals deep connections between SRNG, fixed-length source coding, and resolvability, via equalities to $L_f(\Delta|{\bf X})$ and $R(1-f^{-1}(\Delta)|{\bf X})$, and it applies these results to the rate-distortion perception (RDP) problem, yielding a general lower bound $R_f(D,\Delta) \ge \max\{ r(D|{\bf X}), K_f(\Delta|{\bf X}) \}$ with discussion of conditions for tightness. The paper also presents an alternative expression of the optimum SRNG rate in terms of the smooth max entropy $H_0(\cdot|{\bf X}^n)$, linking SRNG to smooth-entropy frameworks and offering a complementary perspective to information-spectrum methods. Overall, the results deepen the theoretical understanding of SRNG under general divergences and provide practical tools for analyzing perceptual-rate tradeoffs in lossy coding.

Abstract

The self-random number generation (SRNG) problem is considered for general setting. In the literature, the optimum SRNG rate with respect to the variational distance has been discussed. In this paper, we first try to characterize the optimum SRNG rate with respect to a subclass of $f$-divergences. The subclass of $f$-divergences considered in this paper includes typical distance measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence our result can be considered as a generalization of the previous result with respect to the variational distance. Next, we consider the obtained optimum SRNG rate from several viewpoints. The $\varepsilon$-source coding problem is one of related problems with the SRNG problem. Our results reveal how the SRNG problem with the $f$-divergence relate to the $\varepsilon$-fixed-length source coding problem. We also apply our results to the rate distortion perception (RDP) function. As a result, we can establish a lower bound for the RDP function with respect to $f$-divergences using our findings. Finally, we discuss the representation of the optimum SRNG rate using the smooth Rényi entropy.

Optimum Self Random Number Generation Rate and Its Application to Rate Distortion Perception Function

TL;DR

This work extends SRNG from the variational distance to a broad class of -divergences, providing a general rate formula for and deriving explicit expressions for key divergences (e.g., variational distance, reverse KL, Hellinger, and -divergence). It reveals deep connections between SRNG, fixed-length source coding, and resolvability, via equalities to and , and it applies these results to the rate-distortion perception (RDP) problem, yielding a general lower bound with discussion of conditions for tightness. The paper also presents an alternative expression of the optimum SRNG rate in terms of the smooth max entropy , linking SRNG to smooth-entropy frameworks and offering a complementary perspective to information-spectrum methods. Overall, the results deepen the theoretical understanding of SRNG under general divergences and provide practical tools for analyzing perceptual-rate tradeoffs in lossy coding.

Abstract

The self-random number generation (SRNG) problem is considered for general setting. In the literature, the optimum SRNG rate with respect to the variational distance has been discussed. In this paper, we first try to characterize the optimum SRNG rate with respect to a subclass of -divergences. The subclass of -divergences considered in this paper includes typical distance measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence our result can be considered as a generalization of the previous result with respect to the variational distance. Next, we consider the obtained optimum SRNG rate from several viewpoints. The -source coding problem is one of related problems with the SRNG problem. Our results reveal how the SRNG problem with the -divergence relate to the -fixed-length source coding problem. We also apply our results to the rate distortion perception (RDP) function. As a result, we can establish a lower bound for the RDP function with respect to -divergences using our findings. Finally, we discuss the representation of the optimum SRNG rate using the smooth Rényi entropy.
Paper Structure (17 sections, 18 theorems, 81 equations)

This paper contains 17 sections, 18 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental lemmas
  4. Optimum SRNG Rate
  5. General formula
  6. Particulalization
  7. Variational distance
  8. Reverse KL divergence
  9. Hellinger distance
  10. $E_\gamma$-divergence
  11. Discussion
  12. Application to Rate distortion perception problem
  13. Alternative Expression of Optimum SRNG Rate using Smooth Rényi entropy
  14. Concluding Remarks
  15. Proof of Lemma \ref{['lemma:direct2']}
  16. ...and 2 more sections

Key Result

Theorem 2.1

where $\overline{H}({\bf X})$ is called the spectral sup-entropy rate of the source ${\bf X}$Han defined as

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2: Optimum SRNG rate with VD
  • Theorem 2.1: Han Han
  • Definition 2.3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Remark 3.1
  • Lemma 3.2
  • ...and 27 more