[Paper Review] Distributed Source Coding for Correlated Memoryless Gaussian Sources

Yasutada Oohama|arXiv (Cornell University)|Aug 27, 2009

Wireless Communication Security Techniques18 references3 citations

TL;DR

This paper establishes explicit inner and outer bounds for the rate distortion region in a distributed source coding system with L correlated memoryless Gaussian sources, where each source is a noisy observation of a K-dimensional Gaussian source vector. The key contribution is a tight characterization of the sum distortion criterion region and a novel duality that links remote (indirect) source coding to multiterminal (direct) source coding, enabling new partial solutions to the long-standing Gaussian CEO problem.

ABSTRACT

We consider a distributed source coding problem of $L$ correlated Gaussian observations $Y_i, i=1,2,...,L$. We assume that the random vector $Y^{L}={}^{ m t} (Y_1,Y_2,$ $...,Y_L)$ is an observation of the Gaussian random vector $X^K={}^{ m t}(X_1,X_2,...,X_K)$, having the form $Y^L=AX^K+N^L ,$ where $A$ is a $L imes K$ matrix and $N^L={}^{ m t}(N_1,N_2,...,N_L)$ is a vector of $L$ independent Gaussian random variables also independent of $X^K$. The estimation error on $X^K$ is measured by the distortion covariance matrix. The rate distortion region is defined by a set of all rate vectors for which the estimation error is upper bounded by an arbitrary prescribed covariance matrix in the meaning of positive semi definite. In this paper we derive explicit outer and inner bounds of the rate distortion region. This result provides a useful tool to study the direct and indirect source coding problems on this Gaussian distributed source coding system, which remain open in general.

Motivation & Objective

• To derive explicit inner and outer bounds for the rate distortion region in a distributed source coding system with L correlated Gaussian observations.
• To address the remote source coding problem where encoders observe noisy versions of a common source vector.
• To establish a duality between indirect (remote) and direct multiterminal source coding problems for Gaussian sources.
• To provide new partial solutions to the quadratic Gaussian CEO problem and related multiterminal rate distortion problems.

Proposed method

• The system models L noisy observations Y^L = A X^K + N^L, where X^K is a K-dimensional zero-mean Gaussian source and N^L is independent Gaussian noise.
• The distortion is measured via the estimation error covariance matrix, with two criteria: vector distortion (diagonal elements bounded) and sum distortion (trace bounded).
• Outer bounds are derived using information-theoretic inequalities, including entropy power inequalities and conditional mutual information decompositions.
• Inner bounds are constructed via structured coding schemes, particularly leveraging Gaussian Mixture and dirty-paper coding-like techniques.
• A key duality is established between the remote source coding problem and the direct multiterminal source coding problem, showing that solutions to one can be transformed into solutions for the other.
• The analysis uses asymptotic entropy and mutual information bounds in the n-letter extension, applying Markov chains and conditional entropy decompositions to derive tight inequalities.

Experimental results

Research questions

• RQ1 What are the explicit inner and outer bounds for the rate distortion region in a distributed source coding system with correlated Gaussian sources and noisy observations?
• RQ2 How does the sum distortion criterion region behave in the remote source coding model, and under what conditions is the bound tight?
• RQ3 Can the solution to the remote source coding problem be transformed into a solution for the direct multiterminal source coding problem?
• RQ4 What is the role of the matrix A in shaping the rate distortion trade-off in this system?
• RQ5 How does the duality between indirect and direct source coding simplify the analysis of the Gaussian CEO problem?

Key findings

• The paper derives explicit outer and inner bounds for both the vector and sum distortion criteria, providing a complete characterization of the rate distortion region in the Gaussian case.
• For the sum distortion criterion, a matching condition is derived under which the inner and outer bounds coincide, implying a tight characterization.
• The duality result shows that any solution to the remote source coding problem can be converted into a solution for the direct multiterminal source coding problem, and vice versa.
• When K=1, the model reduces to the quadratic Gaussian CEO problem, and the derived bounds recover known results from prior work.
• The inner bound is shown to be tight in the case of scalar source (K=1) and sum distortion, matching the known optimal sum rate.
• The analysis reveals that the optimal coding strategy involves a combination of linear estimation and structured quantization, with the mutual information terms in the bounds reflecting the effective signal-to-noise ratio after estimation.