On the R\'enyi Rate-Distortion-Perception Function and Functional Representations

2026-01-17

AI Summary

This work addresses the fundamental trade-off between compression rate, distortion, and perceptual quality—a challenge inadequately captured by classical rate-distortion theory—by extending the rate-distortion-perception (RDP) framework into the Rényi information-theoretic setting. Focusing on Gaussian sources under joint distortion and perception constraints, the study leverages Sibson’s α-mutual information to characterize the corresponding information-theoretic limits. The core contributions include the first establishment of a Rényi generalized strong functional representation lemma, the derivation of a closed-form expression for the Rényi RDP function in the scalar Gaussian case, and the discovery of a phase transition governed by the parameter α: for 0.5 < α < 1, optimal codebooks exhibit heavy-tailed structures, whereas for α > 1, they collapse to finite support. These findings significantly deepen the understanding of the interplay between perception-distortion trade-offs and abrupt changes in coding structure under shared randomness.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements

Abstract

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

Problem

• Rate-Distortion-Perception
• Rényi information
• perception constraint
• functional representation
• information-theoretic limits

Innovation

• Rényi information theory
• Rate-Distortion-Perception
• α-mutual information
• functional representation
• phase transition