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Low-Rate, Low-Distortion Compression with Wasserstein Distortion

Yang Qiu, Aaron B. Wagner

TL;DR

This work introduces Wasserstein distortion as a unified fidelity-realism measure for rate-distortion theory, using a pooling width $\sigma$ to interpolate between pixel-level fidelity and ensemble realism. It proves continuity results showing $D$ converges to fidelity in the $\sigma\to0$ limit and to realism in the $\sigma\to\infty$ limit for ergodic sources, and analyzes the large-$\sigma$ regime to derive two practical achievability schemes and a converse. The two schemes—Independent Realization and Random Permutation—establish scalable $R$ and $D$ tradeoffs with explicit $\sigma$-dependent rates, leading to an asymptotic region for $(\alpha,\beta)$ that characterizes the vanishing-rate, vanishing-distortion behavior. A sandwich argument ties the analysis together by bounding the distortion between tractable proxy distortions, enabling robust transfer of results to general Wasserstein distortions. Overall, the paper lays theoretical foundations for low-rate, low-distortion compression under a perceptually informed distortion metric, with implications for texture-like and human-visual-system-inspired coding.

Abstract

Wasserstein distortion is a one-parameter family of distortion measures that was recently proposed to unify fidelity and realism constraints. After establishing continuity results for Wasserstein in the extreme cases of pure fidelity and pure realism, we prove the first coding theorems for compression under Wasserstein distortion focusing on the regime in which both the rate and the distortion are small.

Low-Rate, Low-Distortion Compression with Wasserstein Distortion

TL;DR

This work introduces Wasserstein distortion as a unified fidelity-realism measure for rate-distortion theory, using a pooling width to interpolate between pixel-level fidelity and ensemble realism. It proves continuity results showing converges to fidelity in the limit and to realism in the limit for ergodic sources, and analyzes the large- regime to derive two practical achievability schemes and a converse. The two schemes—Independent Realization and Random Permutation—establish scalable and tradeoffs with explicit -dependent rates, leading to an asymptotic region for that characterizes the vanishing-rate, vanishing-distortion behavior. A sandwich argument ties the analysis together by bounding the distortion between tractable proxy distortions, enabling robust transfer of results to general Wasserstein distortions. Overall, the paper lays theoretical foundations for low-rate, low-distortion compression under a perceptually informed distortion metric, with implications for texture-like and human-visual-system-inspired coding.

Abstract

Wasserstein distortion is a one-parameter family of distortion measures that was recently proposed to unify fidelity and realism constraints. After establishing continuity results for Wasserstein in the extreme cases of pure fidelity and pure realism, we prove the first coding theorems for compression under Wasserstein distortion focusing on the regime in which both the rate and the distortion are small.
Paper Structure (10 sections, 7 theorems, 95 equations, 3 figures)

This paper contains 10 sections, 7 theorems, 95 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Wasserstein distortion
  3. Fidelity and Realism as Extreme Cases
  4. Low-rate Scheme Analysis
  5. Achievability -- Two Schemes
  6. Independent Realization
  7. Random Permutation
  8. Converse -- Lower Bound
  9. Sandwich Argument
  10. Proofs for Section \ref{['sec:extreme']}

Key Result

Theorem 3.1

Suppose $q$ satisfies TSG:delta -- TSG:tail and $\mathbf{z}$, $\hat{\mathbf{z}}$, and $q$ together satisfy for all $\sigma > 0$. Then we have

Figures (3)

  • Figure 1: Receptive fields in the ventral stream grow with eccentricity.
  • Figure 2: A pictorial illustration of (\ref{['eq:measure']}) qiu2023wasserstein. In the right plot, the size of the disk indicates the probability mass and the vertical coordinate of the center of the disk indicates the value.
  • Figure 3: Upper- and Lower- bounds for low-rate scheme.

Theorems & Definitions (17)

  • Theorem 3.1
  • Theorem 3.2
  • Corollary
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 7 more