A Revisit to Rate-distortion Theory via Optimal Weak Transport

TL;DR

This paper explores the rate-distortion theory through the lens of optimal weak transport, establishing new connections with Schr"odinger bridge problems and clarifying conditions for optimality and achievability.

Contribution

It extends rate-distortion theory to abstract alphabets using optimal weak transport, providing a parametric representation and linking it to Schr"odinger bridge problems.

Findings

Derived a parametric form of the rate-distortion function.

Connected rate-distortion with Schr"odinger bridge problems.

Reproduced Shannon lower bound achievability without variational calculus.

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 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.