On the Rényi Rate-Distortion-Perception Function and Functional Representations
This work provides theoretical insights into compression under perception constraints for information theorists, but the results are limited to scalar Gaussian sources and specific Rényi parameters.
The paper extends the Rate-Distortion-Perception (RDP) framework to Rényi information theory, deriving closed-form expressions for scalar Gaussian sources and establishing a Rényi-generalized Strong Functional Representation Lemma. It reveals a phase transition in optimal functional representations depending on the Rényi parameter α.
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.