Relaxed Wyner's Common Information

In the problem of coded caching for media delivery, two separate coding opportunities have been identified. The first opportunity is a multi-user advantage and crucially hinges on a public broadcast link in the delivery phase. This has been explored in a plethora of works. The second opportunity has received far less attention and concerns similarities between files in the database. Here, the paradigm is to cache "the similarity" between the files. Upon the request, the encoder refines this by providing the specific details for the requested files. Extending Gray and Wyner's work (1974), it follows that the right measure of file similarity is Wyner's Common Information and its generalizations. The present paper surveys and extends the role of Wyner's Common Information in caching. As a novel result, explicit solutions are found for the Gaussian case under mean-squared error, both for the caching problem as well as for the network considered by Gray and Wyner. Our solution leverages and extends the recent technique of factorization of convex envelopes.