Rate-distortion Theory with Lower Semi-continuous Distortion on Noncompact Alphabets

TL;DR

This paper extends rate-distortion theory to noncompact alphabets with lower semi-continuous distortions, establishing existence of optimal solutions via new mathematical techniques.

Contribution

It introduces two novel existence mechanisms for optimal reconstructions on noncompact alphabets, broadening classical results.

Findings

Optimal reconstruction distributions exist under lower semi-continuity on noncompact alphabets.

For bounded distortions, existence is proved using one-point compactification.

For unbounded coercive distortions, existence is shown via concentration-compactness.

Abstract

In this paper, we study rate-distortion theory for general sources with an emphasis on the existence of optimal reconstruction distributions on noncompact alphabets. Classical attainability results typically rely on compactness of the reproduction alphabet together with continuity of the distortion function, which may fail in many noncompact settings. We identify two complementary existence mechanisms under lower semi-continuity on locally compact Polish alphabets. For bounded distortions, we prove that the rate-distortion infimum is attained via the one-point compactification argument. For unbounded coercive distortions, we establish existence via concentration-compactness. We also give several counterexamples showing that our attainability results are close to sharp. Our results provide a unified and transparent existence theorem for rate-distortion problems with lower semi-continuous distortions.