Generalizing the Posterior Matching Scheme to higher dimensions via optimal transportation

This paper re-visits Shayevitz & Feder´s recent ´Posterior Matching Scheme´, an explicit, dynamical system encoder for communication with feedback that treats the message as a point on the [0, 1] line and achieves capacity on memo-ryless channels. It has two key properties that ensure that it maximizes mutual information at each step: (a) the encoder sequentially hands the decoder what is missing; and (b) the next input has the desired statistics. Motivated by brain-machine interface applications and multi-antenna communications, we consider developing dynamical system feedback encoders for scenarios when the message point lies in higher dimensions. We develop a necessary and sufficient condition the Jacobian equation for any dynamical system encoder that maximizes mutual information. In general, there are many solutions to this equation. We connect this to the Monge-Kantorovich Optimal Transportation Problem, which provides a framework to identify a unique solution suiting a specific purpose. We provide two examplar capacity-achieving solutions for different purposes for the multi-antenna Gaussian channel with feedback. This insight further elucidates an interesting relationship between interactive decision theory problems and the theory of optimal transportation.