Exact Redundancy for Symmetric Rate-Distortion
Sharang M. Sriramu, Aaron B. Wagner
TL;DR
The paper studies exact redundancy for variable-length coding under distortion for symmetric source–distortion pairs, notably a uniform source with permutation-symmetric distortion. It proves that for $0<D<D^*$, the redundancy overhead is precisely $\frac{\log n}{2n}$ for both average-distortion and $d$-semifaithful settings, by combining symmetry-based channel simulation with large-deviations analysis. The achievability leverages a distortion-ball sampling scheme whose index is entropy-coded due to the known distribution, while the converse uses distortion-ball arguments and RD curvature to show the same $\frac{\log n}{2n}$ penalty is unavoidable. These results tighten the understanding of finite-blocklength redundancy in rate-distortion and show that symmetry can yield exact, nontrivial reductions beyond prior bounds, with implications for optimal coding in symmetric settings.
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.