A Revisit to Rate-distortion Problems via Optimal Weak Transport Theory

2025-01-16

AI Summary

This work addresses the rate-distortion optimization problem for complex symbolic systems—particularly abstract alphabets—by introducing a novel framework grounded in optimal weak optimal transport theory. Methodologically, it pioneers the integration of weak transport systems into rate-distortion analysis, enabling unified treatment of discrete, continuous, and abstract settings while circumventing reliance on traditional variational methods; it further synthesizes Schrödinger bridge theory, probabilistic metric space analysis, and convex optimization to derive an explicit parametric characterization of the rate-distortion function. Key contributions include: (i) establishing existence and optimality criteria for solutions in abstract spaces; (ii) concisely recovering and generalizing Rose’s result on achievability of the Shannon lower bound; (iii) uncovering a fundamental connection between rate-distortion theory and the Schrödinger bridge problem; and (iv) proposing a new paradigm bridging information theory and optimal transport.

Technology Category

• Machine Learning: Information Theory
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

This paper revisits the rate-distortion theory from the perspective of optimal weak transport theory, 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 Schr""odinger 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.

Problem

• Information Transmission
• Rate-Distortion Theory
• Complex Symbol Systems

Innovation

• Optimal Weak Transport Theory
• Rate-Distortion Relationship
• Schrödinger Bridge Problem