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The Rate-Distortion-Perception Trade-off: The Role of Private Randomness

Yassine Hamdi, Aaron B. Wagner, Deniz Gündüz

TL;DR

The paper addresses the rate-distortion-perception trade-off for memoryless sources under realism constraints, focusing on the role of private randomness at the encoder. It develops a soft covering lemma for randomized compressors and leverages channel-synthesis techniques to characterize how rate, common randomness, and both encoder/decoder private randomness affect distortion and realism. The main result is a precise, single-letter region $\mathcal{S}_D$ (and its infinite-entropy extension $\mathcal{S}_G$) that captures the five-way trade-off, with encoder private randomness shown to be unnecessary for $R < H(X)$ regardless of other randomness resources. Per-symbol realism strengthens these conclusions, and the achievability uses modifications of standard code constructions together with local channel-synthesis arguments. These findings clarify when local randomness can be ignored and how shared randomness interacts with realism constraints, providing a principled foundation for perception-aware compression and channel-synthesis connections.

Abstract

In image compression, with recent advances in generative modeling, the existence of a trade-off between the rate and the perceptual quality (realism) has been brought to light, where the realism is measured by the closeness of the output distribution to the source. It has been shown that randomized codes can be strictly better under a number of formulations. In particular, the role of common randomness has been well studied. We elucidate the role of private randomness in the compression of a memoryless source $X^n=(X_1,...,X_n)$ under two kinds of realism constraints. The near-perfect realism constraint requires the joint distribution of output symbols $(Y_1,...,Y_n)$ to be arbitrarily close the distribution of the source in total variation distance (TVD). The per-symbol near-perfect realism constraint requires that the TVD between the distribution of output symbol $Y_t$ and the source distribution be arbitrarily small, uniformly in the index $t.$ We characterize the corresponding asymptotic rate-distortion trade-off and show that encoder private randomness is not useful if the compression rate is lower than the entropy of the source, however limited the resources in terms of common randomness and decoder private randomness may be.

The Rate-Distortion-Perception Trade-off: The Role of Private Randomness

TL;DR

The paper addresses the rate-distortion-perception trade-off for memoryless sources under realism constraints, focusing on the role of private randomness at the encoder. It develops a soft covering lemma for randomized compressors and leverages channel-synthesis techniques to characterize how rate, common randomness, and both encoder/decoder private randomness affect distortion and realism. The main result is a precise, single-letter region (and its infinite-entropy extension ) that captures the five-way trade-off, with encoder private randomness shown to be unnecessary for regardless of other randomness resources. Per-symbol realism strengthens these conclusions, and the achievability uses modifications of standard code constructions together with local channel-synthesis arguments. These findings clarify when local randomness can be ignored and how shared randomness interacts with realism constraints, providing a principled foundation for perception-aware compression and channel-synthesis connections.

Abstract

In image compression, with recent advances in generative modeling, the existence of a trade-off between the rate and the perceptual quality (realism) has been brought to light, where the realism is measured by the closeness of the output distribution to the source. It has been shown that randomized codes can be strictly better under a number of formulations. In particular, the role of common randomness has been well studied. We elucidate the role of private randomness in the compression of a memoryless source under two kinds of realism constraints. The near-perfect realism constraint requires the joint distribution of output symbols to be arbitrarily close the distribution of the source in total variation distance (TVD). The per-symbol near-perfect realism constraint requires that the TVD between the distribution of output symbol and the source distribution be arbitrarily small, uniformly in the index We characterize the corresponding asymptotic rate-distortion trade-off and show that encoder private randomness is not useful if the compression rate is lower than the entropy of the source, however limited the resources in terms of common randomness and decoder private randomness may be.
Paper Structure (32 sections, 15 theorems, 13 equations, 1 figure)

This paper contains 32 sections, 15 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Notation
  4. Definitions
  5. A soft covering lemma for randomized compressors
  6. Statement
  7. The rate-distortion-perception trade-off with near-perfect realism
  8. The role of encoder private randomness for finite source alphabets
  9. An extension to sources with infinite entropy
  10. Per-symbol realism
  11. Achievability proof for Theorem \ref{['thm:region_discrete_alphabets']}
  12. Modifying a standard code construction
  13. Achievability of Theorem \ref{['thm:region_discrete_alphabets']}
  14. Conclusion
  15. Some lemmas on the total variation distance
  16. ...and 17 more sections

Key Result

Proposition 4

Consider a finite input alphabet $\mathcal{X},$ a distribution $p_X$ on $\mathcal{X},$ non-negative reals $R,R_c$ and a sequence $\{F^{(n)}\}_{n \geq 1}$ of encoders corresponding to a sequence of $(n,R,R_c,\infty)$ fixed-length codes. The $n$-th induced distribution is denoted $P^{(n)}.$ If $R<H(X)

Figures (1)

  • Figure 1: The system model.

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Remark 5
  • Theorem 6
  • Definition 7
  • Definition 8
  • Claim 9
  • Theorem 10
  • ...and 14 more