Exact Redundancy for Symmetric Rate-Distortion

TL;DR

This paper establishes the exact redundancy bounds for symmetric rate-distortion coding, showing the bounds are tight and achievable for uniform sources under certain symmetry conditions.

Contribution

It proves that the redundancy bounds of log n/n are tight and achievable for symmetric sources, extending previous bounds to exact asymptotic characterization.

Findings

Redundancy bounds are tight and achievable for symmetric sources.

The log n/(2n) redundancy is optimal even with relaxed distortion constraints.

Results apply specifically to uniform sources with symmetric distortion measures.

Abstract

For variable-length coding with an almost-sure distortion constraint, Zhang et al. show that for discrete sources the redundancy is upper bounded by $\log n/n$ and lower bounded (in most cases) by $\log n/(2n)$, ignoring lower order terms. For a uniform source with a distortion measure satisfying certain symmetry conditions, we show that $\log n/(2n)$ is achievable and that this cannot be improved even if one relaxes the distortion constraint to be in expectation rather than with probability one.