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Rate-Distortion-Perception Tradeoff for the Gray-Wyner Problem

Yu Yang, Yingxin Zhang, Weijie Yuan, Lin Zhou

TL;DR

This work extends rate-distortion-perception (RDP) theory to the Gray-Wyner network with two correlated sources by deriving the first-order RDP region. The optimal trade-off is characterized by $R_0 \ge I(X,Y;W)$ for the common message and $R_1 \ge R_{X|W}(Q_{XW},D_1,P_1)$, $R_2 \ge R_{Y|W}(Q_{YW},D_2,P_2)$ for the private messages, optimized over $Q_{XYW}$ with $|\\mathcal{W}|\le |\\mathcal{X}||\\mathcal{Y}|+2$. A random circular shift with common randomness enforces the perceptual constraints, and the authors show that this randomness can be removed via source simulation, yielding deterministic codes. The results preserve the Gray-Wyner separation structure under perceptual constraints and connect to point-to-point RDP and successive-refinement concepts, with implications for distributional perceptual quality in distributed compression systems. The paper thus provides a rigorous, transferable foundation for perceptual-aware multi-terminal lossy compression and informs practical design of distributed image/video codecs.

Abstract

We revisit the Gray-Wyner lossy source coding problem and derive the first-order asymptotic optimal rate-distortion-perception region when additional perception constraints are imposed on reproduced source sequences. The optimal trade-off is shown to be governed by a mutual information term involving common information and two conditional rate-distortion-perception functions. The perception constraint requires that the distribution of each reproduced sequence is close to that of the original source sequence, which is motivated by practical applications in image and video compression. Prior studies usually focus on the compression and reconstruction of a single source sequence. In this paper, we generalize the prior results for point-to-point systems to the representative multi-terminal setting of the Gray-Wyner problem with two correlated source sequences. In particular, we integrate the analyses of the distortion and the perception constraints by including the random circular shift operator in the encoding and decoding process directly.

Rate-Distortion-Perception Tradeoff for the Gray-Wyner Problem

TL;DR

This work extends rate-distortion-perception (RDP) theory to the Gray-Wyner network with two correlated sources by deriving the first-order RDP region. The optimal trade-off is characterized by for the common message and , for the private messages, optimized over with . A random circular shift with common randomness enforces the perceptual constraints, and the authors show that this randomness can be removed via source simulation, yielding deterministic codes. The results preserve the Gray-Wyner separation structure under perceptual constraints and connect to point-to-point RDP and successive-refinement concepts, with implications for distributional perceptual quality in distributed compression systems. The paper thus provides a rigorous, transferable foundation for perceptual-aware multi-terminal lossy compression and informs practical design of distributed image/video codecs.

Abstract

We revisit the Gray-Wyner lossy source coding problem and derive the first-order asymptotic optimal rate-distortion-perception region when additional perception constraints are imposed on reproduced source sequences. The optimal trade-off is shown to be governed by a mutual information term involving common information and two conditional rate-distortion-perception functions. The perception constraint requires that the distribution of each reproduced sequence is close to that of the original source sequence, which is motivated by practical applications in image and video compression. Prior studies usually focus on the compression and reconstruction of a single source sequence. In this paper, we generalize the prior results for point-to-point systems to the representative multi-terminal setting of the Gray-Wyner problem with two correlated source sequences. In particular, we integrate the analyses of the distortion and the perception constraints by including the random circular shift operator in the encoding and decoding process directly.
Paper Structure (22 sections, 2 theorems, 82 equations, 2 figures)

This paper contains 22 sections, 2 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Definitions
  3. Problem Formulation
  4. Definitions
  5. Main Results
  6. Preliminaries
  7. Results and Discussions
  8. Achievability Proof of Theorem \ref{['theorem:1']}
  9. Codebook Generation
  10. Coding Scheme
  11. Distortion Analysis
  12. Perception Analysis
  13. Common Randomness Removal
  14. Conclusion
  15. Proof of First-Order Rate Region (Theorem \ref{['theorem:1']})
  16. ...and 7 more sections

Key Result

Theorem 1

Fix any $(D_1,D_2,P_1,P_2)\in\mathbb{R}_+^4$ and $\dagger\in\{\mathrm{cr,pr,nr}\}$. Under Assumptions assump:1 and assump:2, the RDP region for the Gray-Wyner problem satisfies

Figures (2)

  • Figure 1: Lossy Gray-Wyner model with perception and randomness.
  • Figure 2: Illustration of the encoding process with common randomness.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Lemma 1
  • proof