On the Dispersions of Three Network Information Theory Problems

Details On the Dispersions of Three Network Information Theory Problems On the Dispersions of Three Network Information Theory Problems

2012-01-18

arxiv.org/abs/1201.3901v2

Computer Science

Information Theory

37 pages, 8 figures. Submitted to the IEEE Transactions on Information Theory Version 2: Corrected some minor typos

Scientific paper

We characterize fundamental limits for distributed lossless source coding (the Slepian-Wolf problem), the multiple-access channel and the asymmetric broadcast channel in the finite blocklength setting. For the Slepian-Wolf problem, we introduce a fundamental quantity known as the entropy dispersion matrix, which is analogous to the scalar dispersion quantities that have gained interest in the recent literature. We show that if this matrix is positive-definite, the optimal rate region under the constraint of a fixed blocklength and non-zero error probability has a curved boundary compared to being polyhedral for the asymptotic Slepian-Wolf scenario. In addition, the entropy dispersion matrix governs the rate of convergence of the non-asymptotic region to the asymptotic one. We develop a general universal achievability procedure for finite blocklength analyses of other network information theory problems such as the multiple-access channel and broadcast channel. We provide inner bounds to these problems using a key result known as the vector rate redundancy theorem which is proved using a multidimensional version of the Berry-Esseen theorem. We show that a so-called information dispersion matrix characterizes these inner bounds. Numerical examples show how the non-asymptotic Slepian-Wolf region and multiple-access inner bound compare to their asymptotic counterparts. We also demonstrate numerically that the required blocklengths predicted by dispersion analysis are generally smaller than that predicted by error exponent analysis.

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