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

TL;DR

This paper extends the Rate-Distortion-Perception framework to the Re9nyi information setting, deriving new theoretical results and revealing phase transitions in the complexity of optimal representations for Gaussian sources.

Contribution

It introduces a Re9nyi-based RDP framework, derives closed-form solutions for Gaussian sources, and establishes a generalized Strong Functional Representation Lemma.

Findings

Perception constraint creates a feasible interval for reproduction variance.

For 0.5<b5<1, coding cost is bounded by b5-divergence, requiring heavy-tailed codebooks.

For b5>1, the optimal representation has finite support, simplifying shared randomness compression.

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.