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Rate-Distortion-Perception Theory for the Quadratic Wasserstein Space

Xiqiang Qu, Jun Chen, Lei Yu, Xiangyu Xu

TL;DR

The paper tackles the distortion-rate-perception tradeoff in lossy source coding under squared-error distortion and squared Wasserstein-2 perception, with a cap on common randomness. It derives a unified single-letter characterization $D^*(R,C,P)=D(R,C,P)$, where $D(R,C,P)$ minimizes a sum of a conventional distortion term and a Wasserstein-based perceptual penalty under mutual information constraints, and proves achievability via soft-covering with Wasserstein distance and a linear interpolation decoder. In the Gaussian setting, it provides explicit formulas for scalar and vector sources, showing that a universal representation exists for scalar sources with finite common randomness, while such universality may fail for vector sources when $C>0$. The results unify and extend prior extremes (no randomness, unlimited randomness, perfect perception) and have practical implications for designing perception-aware coding schemes using tools like nested lattice quantization and optimal transport. They also illuminate the role of common randomness in perception-constrained performance and open avenues for extending to other distortion/perception measures and to more general distribution families.

Abstract

We establish a single-letter characterization of the fundamental distortion-rate-perception tradeoff with limited common randomness under the squared error distortion measure and the squared Wasserstein-2 perception measure. Moreover, it is shown that this single-letter characterization can be explicitly evaluated for the Gaussian source. Various notions of universal representation are also clarified.

Rate-Distortion-Perception Theory for the Quadratic Wasserstein Space

TL;DR

The paper tackles the distortion-rate-perception tradeoff in lossy source coding under squared-error distortion and squared Wasserstein-2 perception, with a cap on common randomness. It derives a unified single-letter characterization , where minimizes a sum of a conventional distortion term and a Wasserstein-based perceptual penalty under mutual information constraints, and proves achievability via soft-covering with Wasserstein distance and a linear interpolation decoder. In the Gaussian setting, it provides explicit formulas for scalar and vector sources, showing that a universal representation exists for scalar sources with finite common randomness, while such universality may fail for vector sources when . The results unify and extend prior extremes (no randomness, unlimited randomness, perfect perception) and have practical implications for designing perception-aware coding schemes using tools like nested lattice quantization and optimal transport. They also illuminate the role of common randomness in perception-constrained performance and open avenues for extending to other distortion/perception measures and to more general distribution families.

Abstract

We establish a single-letter characterization of the fundamental distortion-rate-perception tradeoff with limited common randomness under the squared error distortion measure and the squared Wasserstein-2 perception measure. Moreover, it is shown that this single-letter characterization can be explicitly evaluated for the Gaussian source. Various notions of universal representation are also clarified.

Paper Structure

This paper contains 17 sections, 9 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Definition and Literature Review
  3. No Common Randomness $(C=0)$
  4. Unlimited Common Randomness $(C=\infty)$
  5. Perfect Perception $(P=0)$
  6. Coding Theorem
  7. Gaussian Case
  8. Scalar Source
  9. Vector Source
  10. Conclusion
  11. Proof of Proposition \ref{['prop:inf']}
  12. Proof of Proposition \ref{['prop:convexity']}
  13. Proof of Theorem \ref{['thm:main']}
  14. Achievability
  15. Converse
  16. ...and 2 more sections

Key Result

Proposition 1

For $p_X$ with $\mathbb{E}[\|X\|^2]<\infty$, the infimum in eq:inf is attained.

Figures (2)

  • Figure 1: Illustration of $D^*(R,C,P)$ as a function of $R$, for $P=0.1$ and different values of $C$, with $p_X=\mathcal{N}(0,1)$.
  • Figure 2: Illustration of $D^*(R,C,P)$ as a function of $R$, for $C=1$ and different values of $P$, with $p_X=\mathcal{N}(0,1)$.

Theorems & Definitions (10)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4