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Rate-distortion Theory on Non-compact Spaces: A Concentration-compactness Approach

Jiayang Zou, Luyao Fan, Jiayang Gao, Jia Wang

TL;DR

This work addresses the existence of optimal reconstruction distributions for rate-distortion problems on non-compact spaces. It develops a generalized concentration-compactness framework for probability measures on unbounded Polish spaces, coupled with coercivity and lower semi-continuity conditions on the distortion function to guarantee tightness and existence of minimizers without requiring compact reproduction alphabets. The main results include a concentration-compactness lemma adapted to the rate-distortion setting and a coercivity-based existence theorem, which together extend classical results (e.g., Csiszár, Rezaei) to broader, possibly discontinuous distortion functions. The approach unifies and broadens the applicability of rate-distortion theory to general spaces and provides a rigorous variational foundation for optimal lossy coding beyond compact settings.

Abstract

In this paper, we study rate-distortion theory for general sources with an emphasis on the existence of optimal reconstruction distributions. Classical existence results rely on compactness assumptions that are often violated in non-compact settings. By introducing the concentration-compactness principle into the analysis of the rate-distortion functional, we establish the existence of optimal reconstructions under mild coercivity conditions on the distortion function. Our results provide a unified and transparent existence theorem for rate-distortion problems on general non-compact spaces.

Rate-distortion Theory on Non-compact Spaces: A Concentration-compactness Approach

TL;DR

This work addresses the existence of optimal reconstruction distributions for rate-distortion problems on non-compact spaces. It develops a generalized concentration-compactness framework for probability measures on unbounded Polish spaces, coupled with coercivity and lower semi-continuity conditions on the distortion function to guarantee tightness and existence of minimizers without requiring compact reproduction alphabets. The main results include a concentration-compactness lemma adapted to the rate-distortion setting and a coercivity-based existence theorem, which together extend classical results (e.g., Csiszár, Rezaei) to broader, possibly discontinuous distortion functions. The approach unifies and broadens the applicability of rate-distortion theory to general spaces and provides a rigorous variational foundation for optimal lossy coding beyond compact settings.

Abstract

In this paper, we study rate-distortion theory for general sources with an emphasis on the existence of optimal reconstruction distributions. Classical existence results rely on compactness assumptions that are often violated in non-compact settings. By introducing the concentration-compactness principle into the analysis of the rate-distortion functional, we establish the existence of optimal reconstructions under mild coercivity conditions on the distortion function. Our results provide a unified and transparent existence theorem for rate-distortion problems on general non-compact spaces.
Paper Structure (4 sections, 8 theorems, 16 equations)

This paper contains 4 sections, 8 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. A Recap of Rate-Distortion Theory
  3. Main Results
  4. Discussion and Future Work

Key Result

Lemma 1

[Lemma 1.1 in csiszar1974extremum] Under Assumption assumption: distortion function basic requirement, the rate-distortion function $R(D)$ is a non-increasing and convex function of $D$. Moreover, it admits the representation for all $D\leqslant D_{1}$, where $D_1$ denotes the smallest nonnegative number such that $R(D)$ is constant on $(D_1,+\infty)$ (typically corresponding to the threshold bey

Theorems & Definitions (14)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6: Generalized Concentration-compactness Lemma I
  • Remark 1
  • Remark 2
  • Definition 1: Proper Metric Spaces
  • ...and 4 more