R\'enyi Common Information for Doubly Symmetric Binary Sources

Lei Yu

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arXiv · 2407.06132

R\'enyi Common Information for Doubly Symmetric Binary Sources

Lei Yu

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Abstract

In this note, we provide analytic expressions for the R\'enyi common information of orders in $(1,\infty)$ for the doubly symmetric binary source (DSBS). Until now, analytic expressions for the R\'enyi common information of all orders in $[0,\infty]$ have been completely known for this source. We also consider the R\'enyi common information of all orders in $[-\infty,0)$ and evaluate it for the DSBS. We provide a sufficient condition under which the R\'enyi common information of such orders coincides with Wyner's common information for the DSBS. Based on numerical analysis, we conjecture that there is a certain phase transition as the crossover probability increasing for the R\'enyi common information of negative orders for the DSBS. Our proofs are based on a lemma on splitting of the entropy and the analytic expression of relaxed Wyner's common information.

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Lei Yu. 2024-07-08. R\'enyi Common Information for Doubly Symmetric Binary Sources. https://arxiv.org/abs/2407.06132

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