R\'enyi Common Information for Doubly Symmetric Binary Sources
<|reference_start|>R\'enyi Common Information for Doubly Symmetric Binary Sources: 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.<|reference_end|>
@article{yu2024r\'enyi,
    title={R\'enyi Common Information for Doubly Symmetric Binary Sources},
    author={Lei Yu},
    journal={arXiv preprint arXiv:2407.06132},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.06132},
    primaryClass={cs.IT math.IT}
}