  This paper considers an information bottleneck problem with the objective of
obtaining a most informative representation of a hidden feature subject to a
R\'enyi entropy complexity constraint. The optimal bottleneck trade-off between
relevance (measured via Shannon's mutual information) and R\'enyi entropy cost
is defined and an iterative algorithm for finding approximate solutions is
provided. We also derive an operational characterization for the optimal
trade-off by demonstrating that the optimal R\'enyi entropy-relevance trade-off
is achievable by a simple time-sharing scalar coding scheme and that no coding
scheme can provide better performance. Two examples where the optimal Shannon
entropy-relevance trade-off can be exactly determined are further given.
