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Indirect Rate Distortion Functions with Side Information: Structural Properties and Multivariate Gaussian Sources

Evagoras Stylianou, Michail Gkagkos, Charalambos D. Charalambous

TL;DR

A novel realization theory approach is used to establish achievability of the converse coding theorem lower bounds of the two RDFs.

Abstract

In this paper, we analyze the indirect source coding problem with side information at both the encoder and decoder, as well as only at the decoder. We first derive structural properties of the two rate distortion functions (RDFs) for general abstract spaces and identify conditions under which the RDFs coincide. For multivariate jointly Gaussian random variables with square-error fidelity, we establish structural properties of the optimal test channels, show that side information at both the encoder and decoder does not reduce compression, and provide water-filling solutions using parallel Gaussian channel realizations. This paper uses a novel realization theory approach to establish achievability of the converse coding theorem lower bounds of the two RDFs.

Indirect Rate Distortion Functions with Side Information: Structural Properties and Multivariate Gaussian Sources

TL;DR

A novel realization theory approach is used to establish achievability of the converse coding theorem lower bounds of the two RDFs.

Abstract

In this paper, we analyze the indirect source coding problem with side information at both the encoder and decoder, as well as only at the decoder. We first derive structural properties of the two rate distortion functions (RDFs) for general abstract spaces and identify conditions under which the RDFs coincide. For multivariate jointly Gaussian random variables with square-error fidelity, we establish structural properties of the optimal test channels, show that side information at both the encoder and decoder does not reduce compression, and provide water-filling solutions using parallel Gaussian channel realizations. This paper uses a novel realization theory approach to establish achievability of the converse coding theorem lower bounds of the two RDFs.

Paper Structure

This paper contains 10 sections, 12 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement and Discussion
  3. Notation
  4. Main Contribution and Organisation
  5. achievable lower bound on the RDFs
  6. optimal test channel realizations and characterizations of RDFs
  7. Characterization of RDF with Side Information at Encoder and Decoder
  8. Characterization of RDF with Side Information Only at the Decoder
  9. Characterizations of RDFs via Water-Filling Solutions
  10. Conclusion

Key Result

Theorem 1

Suppose that the following conditions hold: (i) $n = n_x=n_s$ and the inverse of $Q_{X,S|Y}= Q_{X,S} - Q_{X,Y}Q_Y^{-1}Q_{Y,S}$ existsthis can be relaxed using Hotelling's canonical variable form hotelling1936relations, (ii) $Q_{S|Y} \succ 0,\; Q_{X|Y} \succ 0$, $Q_{X|Y} \succ Q_{X|S,Y}$. Then defi where $V$ and $U$ are unitary matrices and $D$ contains the singular values of ${\bf Q}$. Then, the

Figures (3)

  • Figure 1: When switch $A$ is closed, side information is available at both the encoder and decoder; when open, it is available only at the decoder.
  • Figure 2: Water-filling solution of $\lambda_i,i=1,\dots,5$ for $\Delta = 42$ of Example 1.
  • Figure 3: The indirect RDF as a function of the distortion $\Delta$ of Example 1.

Theorems & Definitions (29)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Proposition 1
  • proof
  • Lemma 1
  • ...and 19 more