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Many-Help-One Problem for Gaussian Sources with a Tree Structure on their Correlation

Yasutada Oohama

TL;DR

This work analyzes the many-help-one problem for L+1 Gaussian sources under a tree-structured (TS) correlation, extending the CI framework to TS and deriving an explicit outer bound, plus a MI-based sufficient condition for tightness. The authors establish a tight bound on the rate-distortion region under TS when the MI condition holds and provide a recursive, explicit sum-rate formula that recovers the Gaussian CEO result in the appropriate limit. They also relate TS to the binary-tree structure (BTS) and discuss matching conditions, showing that for certain TS cases the full region can be characterized. The results advance the understanding of distributed source coding with structured correlations and have potential implications for practical CEO-type and distributed sensing problems.

Abstract

In this paper we consider the separate coding problem for $L+1$ correlated Gaussian memoryless sources. We deal with the case where $L$ separately encoded data of sources work as side information at the decoder for the reconstruction of the remaining source. The determination problem of the rate distortion region for this system is the so called many-help-one problem and has been known as a highly challenging problem. The author determined the rate distortion region in the case where the $L$ sources working as partial side information are conditionally independent if the remaining source we wish to reconstruct is given. This condition on the correlation is called the CI condition. In this paper we extend the author's previous result to the case where $L+1$ sources satisfy a kind of tree structure on their correlation. We call this tree structure of information sources the TS condition, which contains the CI condition as a special case. In this paper we derive an explicit outer bound of the rate distortion region when information sources satisfy the TS condition. We further derive an explicit sufficient condtion for this outer bound to be tight. In particular, we determine the sum rate part of the rate distortion region for the case where information sources satisfy the TS condition. For some class of Gaussian sources with the TS condition we derive an explicit recursive formula of this sum rate part.

Many-Help-One Problem for Gaussian Sources with a Tree Structure on their Correlation

TL;DR

This work analyzes the many-help-one problem for L+1 Gaussian sources under a tree-structured (TS) correlation, extending the CI framework to TS and deriving an explicit outer bound, plus a MI-based sufficient condition for tightness. The authors establish a tight bound on the rate-distortion region under TS when the MI condition holds and provide a recursive, explicit sum-rate formula that recovers the Gaussian CEO result in the appropriate limit. They also relate TS to the binary-tree structure (BTS) and discuss matching conditions, showing that for certain TS cases the full region can be characterized. The results advance the understanding of distributed source coding with structured correlations and have potential implications for practical CEO-type and distributed sensing problems.

Abstract

In this paper we consider the separate coding problem for correlated Gaussian memoryless sources. We deal with the case where separately encoded data of sources work as side information at the decoder for the reconstruction of the remaining source. The determination problem of the rate distortion region for this system is the so called many-help-one problem and has been known as a highly challenging problem. The author determined the rate distortion region in the case where the sources working as partial side information are conditionally independent if the remaining source we wish to reconstruct is given. This condition on the correlation is called the CI condition. In this paper we extend the author's previous result to the case where sources satisfy a kind of tree structure on their correlation. We call this tree structure of information sources the TS condition, which contains the CI condition as a special case. In this paper we derive an explicit outer bound of the rate distortion region when information sources satisfy the TS condition. We further derive an explicit sufficient condtion for this outer bound to be tight. In particular, we determine the sum rate part of the rate distortion region for the case where information sources satisfy the TS condition. For some class of Gaussian sources with the TS condition we derive an explicit recursive formula of this sum rate part.

Paper Structure

This paper contains 20 sections, 20 theorems, 225 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Problem Statement and Previous results
  3. Formal Statement of the Problem
  4. Tree Structure of Gaussian Sources
  5. Previous Results
  6. Results on the Rate Distortion Region
  7. Definition of Functions and their Properties
  8. Results
  9. Binary Tree Structure Condition
  10. Sum Rate Part of the Rate Distortion Region
  11. Proofs of the Results
  12. Derivation of the Outer Bound
  13. Derivation of the Inner Bound
  14. Proofs of Lemmas \ref{['lm:region1a']} and \ref{['lm:region1b']}
  15. Computation of $\{R_{\rm sum}^{(l)}(D)\}_{l=1}^L$
  16. ...and 5 more sections

Key Result

Theorem 1

For Gaussian sources with general correlation For Gaussian sources with the CI condition the inner bound $\tilde{\cal R}_{L}^{{(\rm in)}}(D)$ is tight, that is

Figures (8)

  • Figure 1: Communication system with $L$ side informations at the decoder.
  • Figure 2: TS condition in the case of $L=4$.
  • Figure 3: CI condition in the case of $L=4$.
  • Figure 4: TS conditions in the case of $L=2$ and the case of $L=3$ and $Z_2=0$.
  • Figure 5: TS condition in the case of $L=3$.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Theorem 1: Oohama oh4
  • Theorem 2
  • Lemma 1
  • Theorem 3
  • Proposition 1
  • Theorem 4: Tavildar et al. tvw
  • Corollary 1: Tavildar et al. tvw
  • Corollary 2
  • Lemma 2
  • Theorem 5
  • ...and 10 more