Gaussian Rate-Distortion-Perception Coding and Entropy-Constrained Scalar Quantization

TL;DR

This work analyzes the quadratic Gaussian rate-distortion-perception problem under limited common randomness for two perception measures: KL divergence and squared Wasserstein-2 distance. It establishes that KL-based and W2-based bounds are nondegenerate (not derivable from one another via refined Talagrand inequalities), and proves an improved, tunable lower bound for the W2 case. By linking rate-distortion-perception to entropy-constrained scalar quantization, it shows that these bounds are not generally tight in the weak-perception regime and that restricting reconstructions to Gaussian distributions can incur penalties under i.i.d. constraints. The results illuminate when Gaussian extremality holds, clarify the relation between KL and W2 measures, and suggest that discrete reconstructions may yield better performance in practice, with important implications for perception-aware coding design.

Abstract

This paper investigates the best known bounds on the quadratic Gaussian distortion-rate-perception function with limited common randomness for the Kullback-Leibler divergence-based perception measure, as well as their counterparts for the squared Wasserstein-2 distance-based perception measure, recently established by Xie et al. These bounds are shown to be nondegenerate in the sense that they cannot be deduced from each other via a refined version of Talagrand's transportation inequality. On the other hand, an improved lower bound is established when the perception measure is given by the squared Wasserstein-2 distance. In addition, it is revealed by exploiting the connection between rate-distortion-perception coding and entropy-constrained scalar quantization that all the aforementioned bounds are generally not tight in the weak perception constraint regime.

Figures (6)

• Figure 1: System diagram.
• Figure 2: Illustrations of $\overline{D}(R,R_c,P|\phi_{KL})$, $\underline{D}(R,R_c,P|\phi_{KL})$, and $\underline{D}(R,R_c,2\sigma^2_X(1-e^{-P})|W^2_2)$ for $p_X=\mathcal{N}(0,1)$, $R_c=0$, and $P=0.1$.
• Figure 3: Illustrations of $\overline{D}(R,R_c,\nu(P)|\phi_{KL})$, $\overline{D}(R,R_c,P|W^2_2)$, and $\underline{D}(R,R_c,P|W^2_2)$ for $p_X=\mathcal{N}(0,1)$, $R_c=0$, and $P=0.1$.
• Figure 4: Illustrations of $\underline{D}'(R,R_c,P|W^2_2)-\underline{D}(R,R_c,P|W^2_2)$ for $p_X=\mathcal{N}(0,1)$, $R=0.1$, and $R_c=0.1$.
• Figure 5: Illustrations of $\underline{D}'(R,R_c,P|W^2_2)-\underline{D}(R,R_c,P|W^2_2)$ for $p_X=\mathcal{N}(0,1)$, $R_c=0.1$, and $P=0.1$.

Theorems & Definitions (21)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9