Exact Redundancy for Symmetric Rate-Distortion

2026-01-17

AI Summary

This study investigates the asymptotic behavior of rate-distortion redundancy for symmetric sources under almost-sure and expected distortion constraints. By leveraging information-theoretic analysis within a variable-length coding framework, the work establishes—for the first time—the exact asymptotic order of the rate-distortion redundancy as $\frac{\log n}{2n}$ for uniform sources satisfying symmetry conditions under symmetric distortion measures. This bound is shown to be both achievable and tight, thereby providing a precise characterization of the fundamental limits. The result delivers a sharp, non-asymptotic benchmark that closes a critical gap in the higher-order asymptotics of rate-distortion theory, offering a complete understanding of redundancy for symmetric sources under both distortion criteria.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• rate-distortion
• redundancy
• symmetric distortion
• variable-length coding
• uniform source

Innovation

• exact redundancy
• symmetric rate-distortion
• almost-sure distortion
• uniform source
• variable-length coding