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Output-Constrained Lossy Source Coding With Application to Rate-Distortion-Perception Theory

Li Xie, Liangyan Li, Jun Chen, Zhongshan Zhang

Abstract

The distortion-rate function of output-constrained lossy source coding with limited common randomness is analyzed for the special case of squared error distortion measure. An explicit expression is obtained when both source and reconstruction distributions are Gaussian. This further leads to a partial characterization of the information-theoretic limit of quadratic Gaussian rate-distortion-perception coding with the perception measure given by Kullback-Leibler divergence or squared quadratic Wasserstein distance.

Output-Constrained Lossy Source Coding With Application to Rate-Distortion-Perception Theory

Abstract

The distortion-rate function of output-constrained lossy source coding with limited common randomness is analyzed for the special case of squared error distortion measure. An explicit expression is obtained when both source and reconstruction distributions are Gaussian. This further leads to a partial characterization of the information-theoretic limit of quadratic Gaussian rate-distortion-perception coding with the perception measure given by Kullback-Leibler divergence or squared quadratic Wasserstein distance.
Paper Structure (11 sections, 12 theorems, 112 equations, 11 figures)

This paper contains 11 sections, 12 theorems, 112 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. General Case
  4. Gaussian Case
  5. Conclusion
  6. Proof of Theorem \ref{['thm:outputconstrained']}
  7. Proof of Corollary \ref{['cor:generallowerbound']}
  8. Proof of Theorem \ref{['thm:Gaussianoutputconstrained']}
  9. Proof of Theorem \ref{['thm:GaussianRDP']}
  10. Proof of Corollary \ref{['cor:Gaussiantight']}
  11. Proof of Theorem \ref{['thm:GaussianW2']}

Key Result

Proposition 1

For $p_{\hat{X}}$ with $\mathbb{E}[\hat{X}^2]<\infty$, we have

Figures (11)

  • Figure 1: System diagram.
  • Figure 2: A generic coding scheme.
  • Figure 3: A specialized coding scheme for squared error distortion measure.
  • Figure 4: Plots of $D(R,R_c|\mathcal{N}(0,1),\mathcal{N}(1,4))$ with $R_c=0, 1, \infty$.
  • Figure 5: Plots of $\overline{D}(R,1,1|\mathcal{N}(0,1))$ and $\underline{D}(R,1,1|\mathcal{N}(0,1))$.
  • ...and 6 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Remark 1
  • Corollary 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • ...and 10 more