Abstract
We revisit the Wyner-Ahlswede-Körner network, focusing especially on the converse part of the dispersion analysis, which is known to be challenging. 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\inf _{ {P}_{ {U}| {X}}}\{{ { cH}}( {Y}| {U})+ {I}( {U}; {X})\}$ </tex-math></inline-formula> with respect to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {Q}_{{{ XY}}}$ </tex-math></inline-formula> , 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {c}\imath _{ {Y}| {U}}( {Y}| {U})+\imath _{ {U}; {X}}( {U}; {X})$ </tex-math></inline-formula> , which improves the existing upper bounds but has a gap to the aforementioned lower bound. Our converse analysis should be immediately extendable to other distributed source-type problems, such as distributed source coding, common randomness generation, and hypothesis testing with communication constraints. We further present improved bounds for the general image-size problem via our semigroup technique.
Highlights
I N THE Wyner-Ahlswede-Körner (WAK) problem [1], [2], [49] (Figure 1), a source Y n and a side information Xn areÔ Õ compressed separately as integers W2 W2 Y n and W1 Ô Õ W1 Xn, respectively, and a decoder reconstructs Yn based onW1 and W2
This article was presented in part at the 2018 IEEE International Symposium on Information Theory (ISIT)
The converse technique in [30] is based on functional-entropic duality and the reverse hypercontractivity of semigroups, which is widely applicable to multiuser information theory problems
Summary
I N THE Wyner-Ahlswede-Körner (WAK) problem [1], [2], [49] (Figure 1), a source Y n and a side information Xn are. The converse technique in [30] is based on functional-entropic duality and the reverse hypercontractivity of semigroups, which is widely applicable to multiuser information theory problems. It appears that all previous strong converses via BUL or the image-size characterization in [1],. Apart from reverse hypercontractivity and functionalentropic duality, another interesting ingredient in our proof is an argument in analyzing certain information quantity for the equiprobable distribution on a type class (Lemma 2) We need this because, unlike Glauber dynamics, the standard tensorization argument for the reverse hypercontractivity (see e.g., [33]) does not apply to the transposition semigroup, and an interesting induction argument is used instead. Ü Ý probability simplex ΔX , the gradient ∇ φ Q can be regarded as a function on X , and ∇ φ Q , Q : ∇ φ Q dQ
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