Dynamical systems, ergodicity, and posterior matching

摘要

The Posterior Matching (PM) scheme is a mutual information maximizing scheme for efficiently communicating a message point in a continuum over a noisy channel with causal feedback. It was originally developed when the message point was on a subset of the real line, and we more recently generalized the framework to arbitrary dimensions with optimal trasnsport theory. Here, we consider a class of encoder dynamical systems constructed with optimal transport and show that the ergodicity of the aforementioned state variable is a necessary and sufficient condition for posterior matching to be reliable. Lastly, we show a surprising “all or nothing” result: this same ergodicity condition is necessary and sufficient to achieve capacity.