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On the Rényi Rate-Distortion-Perception Function and Functional Representations

Jiahui Wei, Marios Kountouris

TL;DR

This work generalizes Rate-Distortion-Perception (RDP) theory to the Rényi information setting by leveraging Sibson's $α$-mutual information. It derives a closed-form Rényi RDP for scalar Gaussian sources and proves a Rényi-generalized Strong Functional Representation Lemma via a Poisson functional representation, uncovering a phase transition in optimal representations: heavy-tailed, effectively infinite codebooks for $0.5<α<1$ versus finite-support representations for $α>1$. The Gaussian case is paired with a perception constraint that induces feasibility intervals on reproduction variance, and a projection-based optimality description is given. Numerical simulations corroborate the theory, and connections to Campbell's coding cost provide an operational interpretation of the resulting bounds. Overall, the paper establishes fundamental limits and constructive schemes for distortion-perception trade-offs under generalized Rényi measures, with implications for robust and distribution-aware compression and generative modeling.

Abstract

We extend the Rate-Distortion-Perception (RDP) framework to the Rényi information-theoretic regime, utilizing Sibson's $α$-mutual information to characterize the fundamental limits under distortion and perception constraints. For scalar Gaussian sources, we derive closed-form expressions for the Rényi RDP function, showing that the perception constraint induces a feasible interval for the reproduction variance. Furthermore, we establish a Rényi-generalized version of the Strong Functional Representation Lemma. Our analysis reveals a phase transition in the complexity of optimal functional representations: for $0.5<α< 1$, the coding cost is bounded by the $α$-divergence of order $α+1$, necessitating a codebook with heavy-tailed polynomial decay; conversely, for $α> 1$, the representation collapses to one with finite support, offering new insights into the compression of shared randomness under generalized notions of mutual information.

On the Rényi Rate-Distortion-Perception Function and Functional Representations

TL;DR

This work generalizes Rate-Distortion-Perception (RDP) theory to the Rényi information setting by leveraging Sibson's -mutual information. It derives a closed-form Rényi RDP for scalar Gaussian sources and proves a Rényi-generalized Strong Functional Representation Lemma via a Poisson functional representation, uncovering a phase transition in optimal representations: heavy-tailed, effectively infinite codebooks for versus finite-support representations for . The Gaussian case is paired with a perception constraint that induces feasibility intervals on reproduction variance, and a projection-based optimality description is given. Numerical simulations corroborate the theory, and connections to Campbell's coding cost provide an operational interpretation of the resulting bounds. Overall, the paper establishes fundamental limits and constructive schemes for distortion-perception trade-offs under generalized Rényi measures, with implications for robust and distribution-aware compression and generative modeling.

Abstract

We extend the Rate-Distortion-Perception (RDP) framework to the Rényi information-theoretic regime, utilizing Sibson's -mutual information to characterize the fundamental limits under distortion and perception constraints. For scalar Gaussian sources, we derive closed-form expressions for the Rényi RDP function, showing that the perception constraint induces a feasible interval for the reproduction variance. Furthermore, we establish a Rényi-generalized version of the Strong Functional Representation Lemma. Our analysis reveals a phase transition in the complexity of optimal functional representations: for , the coding cost is bounded by the -divergence of order , necessitating a codebook with heavy-tailed polynomial decay; conversely, for , the representation collapses to one with finite support, offering new insights into the compression of shared randomness under generalized notions of mutual information.
Paper Structure (22 sections, 5 theorems, 62 equations, 2 figures)

This paper contains 22 sections, 5 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Rényi information quantities
  5. Rate-distortion and rate-distortion-perception theory
  6. Rényi Rate-Distortion-Perception Function
  7. Problem formulation
  8. Optimization properties and convexity
  9. Case study: Gaussian R-RDP function
  10. The Gaussian solution
  11. Incorporating perception constraint
  12. Rényi-Strong Functional Representation Bounds
  13. Proof: Construction via Poisson Functional Representation
  14. Link to Campbell's coding cost
  15. Numerical simulation
  16. ...and 7 more sections

Key Result

Theorem 1

For a Gaussian source $X \sim \mathcal{N}(0, \sigma_X^2)$ and squared -error distortion $D \in (0, \sigma_X^2)$, the Rényi rate-distortion function is given by The optimum is achieved by the standard affine Gaussian test channel $Y = cX + Z$, where $Z\sim \mathcal{N}(0, \sigma_Z^2)$ is Gaussian noise independent of $X$.

Figures (2)

  • Figure 1: Contour plots of the Gaussian R-RDP function
  • Figure 2: Numerical Validation of Rényi-SFRL ($\alpha = 0.6$ and $\alpha = 2$). Left: Histogram of the selected codebook indices $K$ (log scale), showing heavy-tailed behavior for $\alpha < 1$ and finite support for $\alpha > 1$. Right: Convergence of the empirical $\alpha$-moment $\mathbb{E}[K^{0.6}]$ ($\mathbb{E}[\log (K)]$ for $\alpha = 2$, blue), remaining strictly below the theoretical bound (red).

Theorems & Definitions (17)

  • Definition 1: Rényi Entropy renyi1961measures
  • Definition 2: Rényi Divergence renyi1961measures
  • Definition 3: Sibson's $\alpha$-Mutual Information Sibson1969
  • Definition 4: RDP function BlauMichaeliICML2019
  • Remark 1
  • Theorem 1: Gaussian Rényi Rate-Distortion
  • proof : Sketch of Proof
  • Theorem 2: Gaussian R-RDP function
  • proof
  • Remark 2
  • ...and 7 more