Successive Refinement of Abstract Sources

TL;DR

This paper provides a comprehensive parametric characterization of the rate-distortion region for successive refinement of abstract sources, extending existing results and introducing new algorithms and bounds.

Contribution

It extends Csiszar's result to abstract sources, generalizes Tuncel and Rose's finite alphabet result, and develops an iterative algorithm and nonasymptotic bounds for the rate-distortion region.

Findings

Derived a parametric characterization of the rate-distortion region.

Extended existing theoretical results to abstract sources.

Developed an iterative algorithm for computing the rate-distortion region.

Abstract

In successive refinement of information, the decoder refines its representation of the source progressively as it receives more encoded bits. The rate-distortion region of successive refinement describes the minimum rates required to attain the target distortions at each decoding stage. In this paper, we derive a parametric characterization of the rate-distortion region for successive refinement of abstract sources. Our characterization extends Csiszar's result to successive refinement, and generalizes a result by Tuncel and Rose, applicable for finite alphabet sources, to abstract sources. This characterization spawns a family of outer bounds to the rate-distortion region. It also enables an iterative algorithm for computing the rate-distortion region, which generalizes Blahut's algorithm to successive refinement. Finally, it leads a new nonasymptotic converse bound. In all the scenarios where the dispersion is known, this bound is second-order optimal. In our proof technique, we avoid Karush-Kuhn-Tucker conditions of optimality, and we use basic tools of probability theory. We leverage the Donsker-Varadhan lemma for the minimization of relative entropy on abstract probability spaces.