On the Dispersions of Three Network Information Theory Problems

TL;DR

This paper investigates the non-asymptotic behavior of three fundamental network information theory problems, introducing new dispersion matrices and operational measures to quantify rate convergence and losses.

Contribution

It introduces the entropy dispersion matrix for the Slepian-Wolf problem, characterizes local and weighted sum-rate dispersions, and extends dispersion analysis to multiple-access and broadcast channels.

Findings

Entropy dispersion matrix characterizes rate loss at finite blocklengths.

Local dispersion approaches a bivariate Gaussian near boundary points.

Inner bounds for channels are derived using dispersion matrices.

Abstract

We analyze the dispersions of distributed lossless source coding (the Slepian-Wolf problem), the multiple-access channel and the asymmetric broadcast channel. For the two-encoder Slepian-Wolf problem, we introduce a quantity known as the entropy dispersion matrix, which is analogous to the scalar dispersions that have gained interest recently. We prove a global dispersion result that can be expressed in terms of this entropy dispersion matrix and provides intuition on the approximate rate losses at a given blocklength and error probability. To gain better intuition about the rate at which the non-asymptotic rate region converges to the Slepian-Wolf boundary, we define and characterize two operational dispersions: the local dispersion and the weighted sum-rate dispersion. The former represents the rate of convergence to a point on the Slepian-Wolf boundary while the latter represents the fastest rate for which a weighted sum of the two rates converges to its asymptotic fundamental limit. Interestingly, when we approach either of the two corner points, the local dispersion is characterized not by a univariate Gaussian but a bivariate one as well as a subset of off-diagonal elements of the aforementioned entropy dispersion matrix. Finally, we demonstrate the versatility of our achievability proof technique by providing inner bounds for the multiple-access channel and the asymmetric broadcast channel in terms of dispersion matrices. All our proofs are unified a so-called vector rate redundancy theorem which is proved using the multidimensional Berry-Esseen theorem.