Dispersion Bound for the Wyner-Ahlswede-K\"orner Network via Reverse Hypercontractivity on Types

TL;DR

This paper develops a new converse approach for distributed source problems using reverse hypercontractivity, providing lower bounds on error probability and dispersion that improve upon existing bounds for the Wyner-Ahlswede-K"orner network.

Contribution

It introduces a novel converse machinery based on reverse hypercontractivity for bounding error and dispersion in distributed source coding problems.

Findings

Lower bounds on error probability for joint types.

Lower bounds on the c-dispersion as the variance of a specific gradient.

Improved upper bounds on c-dispersion compared to previous methods.

Abstract

This paper introduces a new converse machinery for a challenging class of distributed source-type problems (e.g.\ distributed source coding, common randomness generation, or hypothesis testing with communication constraints), through the example of the Wyner-Ahlswede-K\"orner network. Using the functional-entropic duality and the reverse hypercontractivity of the transposition semigroup, we lower bound the error probability for each joint type. Then by averaging the error probability over types, we lower bound the $c$-dispersion (which characterizes the second-order behavior of the weighted sum of the rates of the two compressors when a nonvanishing error probability is small) as the variance of the gradient of $\inf_{P_{U|X}}\{cH(Y|U)+I(U;X)\}$ with respect to $Q_{XY}$, the per-letter side information and source distribution. In comparison, using standard achievability arguments based on the method of types, we upper-bound the $c$-dispersion as the variance of $c\imath_{Y|U}(Y|U)+\imath_{U;X}(U;X)$, which improves the existing upper bounds but has a gap to the aforementioned lower bound.