An Information Bottleneck Problem with Rényi's Entropy

This paper considers an information bottleneck problem with the objective of obtaining a most informative representation of a hidden feature subject to a Renyi entropy complexity constraint. The optimal bottleneck trade-off between relevance (measured via Shannon's mutual information) and Renyi 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 Renyi 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 tradeoff can be exactly determined are further given.