# Rate-Distortion Region of a Gray-Wyner Model with Side Information

## TL;DR

This paper provides a comprehensive single-letter characterization of the rate-distortion region for a Gray-Wyner model with side information at the decoders, considering correlated sources and multiple fidelity constraints.

## Contribution

It introduces a full single-letter characterization of the rate-distortion region for the Gray-Wyner model with side information, including specializations to Heegard-Berger models.

## Key findings

- Derived the rate-distortion region for the model with side information.
- Analyzed the roles of common and private descriptions in coding.
- Provided binary examples illustrating the theoretical results.

## Abstract

In this work, we establish a full single-letter characterization of the rate-distortion region of an instance of the Gray-Wyner model with side information at the decoders. Specifically, in this model an encoder observes a pair of memoryless, arbitrarily correlated, sources $(S^n_1,S^n_2)$ and communicates with two receivers over an error-free rate-limited link of capacity $R_0$, as well as error-free rate-limited individual links of capacities $R_1$ to the first receiver and $R_2$ to the second receiver. Both receivers reproduce the source component $S^n_2$ losslessly; and Receiver $1$ also reproduces the source component $S^n_1$ lossily, to within some prescribed fidelity level $D_1$. Also, Receiver $1$ and Receiver $2$ are equipped respectively with memoryless side information sequences $Y^n_1$ and $Y^n_2$. Important in this setup, the side information sequences are arbitrarily correlated among them, and with the source pair $(S^n_1,S^n_2)$; and are not assumed to exhibit any particular ordering. Furthermore, by specializing the main result to two Heegard-Berger models with successive refinement and scalable coding, we shed light on the roles of the common and private descriptions that the encoder should produce and what they should carry optimally. We develop intuitions by analyzing the developed single-letter optimal rate-distortion regions of these models, and discuss some insightful binary examples.