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Tight Exponential Strong Converses for Lossy Source Coding with Side-Information and Distributed Function Computation

Shun Watanabe

TL;DR

The paper derives the tight exponential strong converse exponent for lossy source coding with side-information (Wyner-Ziv problem) and, as a special case, distributed function computation. It combines a change-of-measure converse with a soft Markov constraint and a Poisson/exponential matching-based achievability to obtain F(R,D) = F^*(R,D), extending previous WAK results to the more intricate WZ setting. A key insight is that the conditional mutual information form of the WZ rate-distortion function already incorporates the soft Markov constraint, enabling a tight exponent derivation. The work also clarifies the role of two soft Markov constraints in distributed coding and demonstrates their necessity via an AND-function example, with the reversible-direction perspective in achievability aided by Poisson matching.

Abstract

The exponential strong converse for a coding problem states that, if a coding rate is beyond the theoretical limit, the correct probability converges to zero exponentially. For the lossy source coding with side-information, also known as the Wyner-Ziv (WZ) problem, a lower bound on the strong converse exponent was derived by Oohama. In this paper, we derive the tight strong converse exponent for the WZ problem; as a special case, we also derive the tight strong converse exponent for the distributed function computation problem. For the converse part, we use the change-of-measure argument developed in the literature and the soft Markov constraint introduced by Oohama; the matching achievability is proved via the Poisson matching approach recently introduced by Li and Anantharam. Our result is build upon the recently derived tight strong converse exponent for the Wyner-Ahlswede-Korner (WAK) problem; however, compared to the WAK problem, more sophisticated argument is needed. As an illustration of the necessity of the soft Markov constraint, we present an example such that the soft Markov constraint is strictly positive.

Tight Exponential Strong Converses for Lossy Source Coding with Side-Information and Distributed Function Computation

TL;DR

The paper derives the tight exponential strong converse exponent for lossy source coding with side-information (Wyner-Ziv problem) and, as a special case, distributed function computation. It combines a change-of-measure converse with a soft Markov constraint and a Poisson/exponential matching-based achievability to obtain F(R,D) = F^*(R,D), extending previous WAK results to the more intricate WZ setting. A key insight is that the conditional mutual information form of the WZ rate-distortion function already incorporates the soft Markov constraint, enabling a tight exponent derivation. The work also clarifies the role of two soft Markov constraints in distributed coding and demonstrates their necessity via an AND-function example, with the reversible-direction perspective in achievability aided by Poisson matching.

Abstract

The exponential strong converse for a coding problem states that, if a coding rate is beyond the theoretical limit, the correct probability converges to zero exponentially. For the lossy source coding with side-information, also known as the Wyner-Ziv (WZ) problem, a lower bound on the strong converse exponent was derived by Oohama. In this paper, we derive the tight strong converse exponent for the WZ problem; as a special case, we also derive the tight strong converse exponent for the distributed function computation problem. For the converse part, we use the change-of-measure argument developed in the literature and the soft Markov constraint introduced by Oohama; the matching achievability is proved via the Poisson matching approach recently introduced by Li and Anantharam. Our result is build upon the recently derived tight strong converse exponent for the Wyner-Ahlswede-Korner (WAK) problem; however, compared to the WAK problem, more sophisticated argument is needed. As an illustration of the necessity of the soft Markov constraint, we present an example such that the soft Markov constraint is strictly positive.

Paper Structure

This paper contains 6 sections, 6 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation and Results
  3. Proof of Converse
  4. Proof of Achievability
  5. Exponential Matching Lemma
  6. Cardinality bounds

Key Result

Proposition 1

For a given distortion level $D\ge 0$, it holds that

Figures (1)

  • Figure 1: A comparison of \ref{['eq:time-sharing-exponent-AND']} (blue dashed line) and \ref{['eq:coded-exponent-AND']} (red solid line). The horizontal axis is $0 \le R \le 1$ and the vertical axis is the correct probability exponent.

Theorems & Definitions (15)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Corollary 1: Distributed function computation
  • proof
  • Corollary 2: Slepian-Wolf coding
  • proof
  • Example 1: AND function for independent sources
  • Remark 3
  • ...and 5 more