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A Revisit to Rate-distortion Theory via Optimal Weak Transport

Jiayang Zou, Luyao Fan, Jiayang Gao, Jia Wang

TL;DR

Within the framework of weak transport problems, a parametric representation of the rate-distortion function is derived, thereby connecting the rate-distortion function with the Schr\"odinger bridge problem, and establishing necessary conditions for its optimality.

Abstract

This paper revisits the rate-distortion theory from the perspective of optimal weak transport, as recently introduced by Gozlan et al. While the conditions for optimality and the existence of solutions are well-understood in the case of discrete alphabets, the extension to abstract alphabets requires more intricate analysis. Within the framework of weak transport problems, we derive a parametric representation of the rate-distortion function, thereby connecting the rate-distortion function with the Schrödinger bridge problem, and establish necessary conditions for its optimality. As a byproduct of our analysis, we reproduce K. Rose's conclusions regarding the achievability of Shannon lower bound concisely, without reliance on variational calculus.

A Revisit to Rate-distortion Theory via Optimal Weak Transport

TL;DR

Within the framework of weak transport problems, a parametric representation of the rate-distortion function is derived, thereby connecting the rate-distortion function with the Schr\"odinger bridge problem, and establishing necessary conditions for its optimality.

Abstract

This paper revisits the rate-distortion theory from the perspective of optimal weak transport, as recently introduced by Gozlan et al. While the conditions for optimality and the existence of solutions are well-understood in the case of discrete alphabets, the extension to abstract alphabets requires more intricate analysis. Within the framework of weak transport problems, we derive a parametric representation of the rate-distortion function, thereby connecting the rate-distortion function with the Schrödinger bridge problem, and establish necessary conditions for its optimality. As a byproduct of our analysis, we reproduce K. Rose's conclusions regarding the achievability of Shannon lower bound concisely, without reliance on variational calculus.
Paper Structure (13 sections, 15 theorems, 77 equations)

This paper contains 13 sections, 15 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Background and Problem Setup
  3. Fundamentals of Weak Transport Theory
  4. A Sketch of The Schödinger Problem
  5. A Recap of Rate-distortion Theory
  6. OWT Meets Rate-distortion Theory
  7. Discussion and Future Work
  8. Proof of Lemma \ref{['lemma: transform the rate-distortion problem']}
  9. Proof of Lemma \ref{['lemma: lower semicontinuity of C']}
  10. Proof of Theorem \ref{['thm: refined existence']}
  11. Proof of Proposition \ref{['proposition: the target measure has finite t-Wasserstein distance']}
  12. Proof of Theorem \ref{['thm: parametric representation-1']}
  13. Proof of Theorem \ref{['thm: parametric representation-2']}

Key Result

Theorem 1

[Theorem 1.1 in backhoff2019existence] Assume that $C:\mathcal{X}\times\mathcal{P}(\mathcal{Y})\to\mathbb{R}\cup\{+\infty\}$ is jointly lower semicontinuous, bounded from below and convex in the second argument. Then the problem admits a minimizer.

Theorems & Definitions (24)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 3
  • proof
  • ...and 14 more