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This tutorial paper focuses on the variants of the bottleneck problem takingan information theoretic perspective and discusses practical methods to solveit, as well as its connection to coding and learning aspects. The intimateconnections of this setting to remote source-coding under logarithmic lossdistortion measure, information combining, common reconstruction, theWyner-Ahlswede-Korner problem, the efficiency of investment information, aswell as, generalization, variational inference, representation learning,autoencoders, and others are highlighted. We discuss its extension to thedistributed information bottleneck problem with emphasis on the Gaussian modeland highlight the basic connections to the uplink Cloud Radio Access Networks(CRAN) with oblivious processing. For this model, the optimal trade-offsbetween relevance (i.e., information) and complexity (i.e., rates) in thediscrete and vector Gaussian frameworks is determined. In the concludingoutlook, some interesting problems are mentioned such as the characterizationof the optimal inputs ("features") distributions under power limitationsmaximizing the "relevance" for the Gaussian information bottleneck, under"complexity" constraints.

On the Information Bottleneck Problems: Models, Connections, Applications and Information Theoretic Views