Tight Exponential Strong Converse for Source Coding Problem With Encoded Side Information

Abstract

The source coding problem with encoded side information is considered. A lower bound on the strong converse exponent has been derived by Oohama, but its tightness has not been clarified. In this paper, we derive a tight strong converse exponent. For the special case where the side-information does not exist, we demonstrate that our tight exponent of the Wyner-Ahlswede-Körner (WAK) problem reduces to the known tight expression of that special case while Oohama’s lower bound is strictly loose. The converse part is proved by a judicious use of the change-of-measure argument, which was introduced by Gu and Effros and further developed by Tyagi and Watanabe. A key component of the methodology by Tyagi and Watanabe is the use of soft Markov constraint, which was originally introduced by Oohama, as a penalty term to prove the Markov constraint at the end. A technical innovation of this paper compared to Tyagi and Watanabe is recognizing that the soft Markov constraint is a part of the exponent, rather than a penalty term that should vanish at the end; this recognition enables us to derive the matching achievability bound. In fact, via numerical experiment, we provide evidence that the soft Markov constraint is strictly positive. Compared to Oohama’s derivation of the lower bound, which relies on the single-letterization of a certain moment-generating function, the derivation of our tight exponent only involves manipulations of the Kullback-Leibrer divergence and Shannon entropies. The achievability part is derived by a careful analysis of the type argument; however, unlike the conventional analysis for the achievable rate region, we need to derive the soft Markov constraint in the analysis of the correct probability. Furthermore, we present an application of our derivation of the strong converse exponent to the privacy amplification.