R\'enyi Common Information for Doubly Symmetric Binary Sources

TL;DR

This paper derives explicit formulas for the R\'enyi common information of all orders for the doubly symmetric binary source, revealing phase transitions and conditions where it matches Wyner's common information.

Contribution

It provides the first complete analytic expressions for R\'enyi common information of all orders for DSBS, including negative orders, and explores phase transitions.

Findings

Explicit formulas for R\'enyi common information of all orders for DSBS.

Identification of conditions where R\'enyi common information equals Wyner's.

Numerical evidence suggesting phase transitions in negative orders.

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.