One-Shot Coding and Applications
Yanxiao Liu
TL;DR
This thesis develops a comprehensive one-shot information-theoretic framework built on Poisson functional representations to address source and channel coding when blocklength is one and the components may be arbitrary. It introduces a unified one-shot network coding bound for general acyclic networks, a one-shot secrecy analysis for information hiding and compound wiretap channels, and a novel Poisson private representation (PPR) that compresses differential privacy mechanisms with near-optimal size while preserving exact output distributions. The work further applies these ideas to distributed mean estimation, providing both central and local DP guarantees and empirical demonstrations of improved privacy-communication-accuracy trade-offs. Collectively, the results unify one-shot coding with differential privacy, offer practical channel-simulation tools for privacy-aware communication, and open paths toward automated, one-shot network analyses and faster DP-enabled data processing, all while recovering classical asymptotic results in appropriate limits.
Abstract
One-shot information theory addresses scenarios in source coding and channel coding where the signal blocklength is assumed to be 1. In this case, each source and channel can be used only once, and the sources and channels are arbitrary and not required to be memoryless or ergodic. We study the achievability part of one-shot information theory, i.e., we consider explicit coding schemes in the oneshot scenario. The objective is to derive one-shot achievability results that can imply existing (first-order and second-order) asymptotic results when applied to memoryless sources and channels, or applied to systems with memory that behave ergodically. Poisson functional representation was first proposed as a one-shot channel simulation technique by Li and El Gamal [118] for proving a strong functional representation lemma. It was later extended to the Poisson matching lemma by Li and Anantharam [117], which provided a unified one-shot coding scheme for a broad class of information-theoretic problems. The main contribution of this thesis is to extend the applicability of Poisson functional representation to various more complicated scenarios, where the original version cannot be applied directly and further extensions must be developed.