[Paper Review] Source Coding for a Simple Network with Receiver Side Information

Roy Timo, Alex Grant+2 more|ArXiv.org|May 27, 2008

Wireless Communication Security Techniques8 references4 citations

TL;DR

This paper extends the classic Gray-Wyner source coding model by incorporating receiver side information, deriving inner and outer bounds for the achievable rate region. For three special cases—identical sources, degraded networks, and complementary delivery—tight outer bounds are proven, yielding exact characterization of the rate region.

ABSTRACT

We consider the problem of source coding with receiver side information for the simple network proposed by R. Gray and A. Wyner in 1974. In this network, a transmitter must reliably transport the output of two correlated information sources to two receivers using three noiseless channels: a public channel which connects the transmitter to both receivers, and two private channels which connect the transmitter directly to each receiver. We extend Gray and Wyner's original problem by permitting side information to be present at each receiver. We derive inner and outer bounds for the achievable rate region and, for three special cases, we show that the outer bound is tight.

Motivation & Objective

• To characterize the achievable rate region for a two-receiver network with side information at each receiver.
• To extend the classical Gray-Wyner model—originally for correlated sources without side information—by incorporating receiver-side auxiliary random variables.
• To derive general inner and outer bounds on the rate region using information-theoretic techniques.
• To identify and solve three special cases (identical sources, degraded, complementary delivery) where the outer bound becomes tight.
• To provide a complete characterization of the rate region in these special cases, offering exact performance limits.

Proposed method

• Derives an outer bound on the achievable rate region using a Markov chain structure and information inequalities, particularly leveraging conditional mutual information and entropy terms.
• Proposes an inner bound via a coding scheme extending Gray and Wyner’s original two-descriptions approach, incorporating auxiliary random variables to model side information.
• Uses random coding with typical sequences and joint typicality decoding to analyze error probabilities across the three channels (public and two private).
• Applies the Markov chain condition $W o (X,Y) o (U,V)$ to simplify mutual information expressions and derive bounds.
• Employs a minimization over auxiliary random variables $A$ and $B$ (with bounded cardinality) to express the inner bound in a parametric form.
• For special cases, proves equality between the outer and inner bounds by showing that the outer bound is achievable using the proposed coding scheme.

Experimental results

Research questions

• RQ1 What is the achievable rate region for a two-receiver network with correlated sources and side information at both receivers?
• RQ2 How does the presence of side information at receivers alter the optimal decomposition of source data across public and private channels compared to the original Gray-Wyner model?
• RQ3 Under what conditions does the derived outer bound become tight, and when can the rate region be exactly characterized?
• RQ4 Can the coding strategy from Gray and Wyner be extended to handle side information while preserving optimality in specific network configurations?
• RQ5 What are the performance limits in special cases such as identical sources, degraded networks, and complementary delivery?

Key findings

• For the case where $X = Y$, the achievable rate region is exactly characterized as the closure of the union of rate regions defined over auxiliary variables $A$ and $B$, with cardinality constraints $|\mathscr{A}| \leq |\mathscr{X}|+1$ and $|\mathscr{B}| \leq |\mathscr{X}|+1$.
• In the degraded network where $Y = (X,Z)$ and $(X,Z) \to U \to V$ forms a Markov chain, the outer bound is tight, so $\mathscr{R} = \left(\bigcup_{p \in \mathscr{P}} \mathscr{R}_{\text{out}}^{(p)}\right)^c$.
• For the complementary delivery case with $U = Y$ and $V = X$, the outer bound is tight, and the rate region is characterized as $\mathscr{R} = \left(\bigcup_{p \in \mathscr{P}^{**}} \mathscr{R}^{(p)\text{**}}\right)^c$, with $|\mathscr{A}| \leq |\mathscr{X}||\mathscr{Y}|+1$ and $|\mathscr{B}| \leq |\mathscr{X}||\mathscr{Y}|+1$.
• The outer bound is derived using information inequalities and is shown to be tight in three distinct network configurations, providing exact performance limits.
• The inner bound is constructed using a modified Gray-Wyner coding scheme with auxiliary random variables, and error probability analysis confirms reliability under the derived rate constraints.
• The results show that side information at receivers fundamentally changes the optimal coding strategy, and the public channel is no longer used solely for common information but must account for side information correlation.