AI summaryⓘ
The authors examine how to best send a real number through a channel with more than one use, while getting feedback. They find conditions that allow the sender to maximize the shared information between the source and all outputs, and this is especially clear for common channel types. They also solve a problem where the goal is not just to maximize information but also to minimize the estimation error at each step, focusing on certain symmetric discrete channels. Their results show that a known method called posterior matching is essentially the only way to achieve both goals for these channels. Overall, the authors offer a new way to balance minimizing error while maximizing information in feedback communication.
Mutual InformationSource CodingFeedback ChannelMMSE (Minimum Mean Square Error)Posterior Matching SchemeDiscrete Channelsk-ary Symmetric Channelk-ary Erasure ChannelInformation MaximizationDistortion Minimization
Abstract
We study the problem of optimally sending a real-valued source through multiple uses of a channel with feedback. First, we state a set of conditions that are sufficient for an encoder to achieve maximal mutual information between the source and all the channel outputs. This set of conditions are also necessary when the channel is input-identifiable, a condition widely satisfied by common channel models. More notably, we further study the information maximization-distortion minimization problem, where the mutual information between the source and all channel outputs still needs to be maximized, while at each step, the MMSE of estimating the source from the channel outputs so far also needs to be minimized. We derive a solution to this problem for discrete channels with certain symmetries, e.g. $k$-ary symmetric or $k$-ary erasure channels. We show that for such channels the famous posterior matching scheme, while not necessary for information maximization alone, is sufficient and essentially necessary for achieving both information maximization and distortion minimization. This work also provides a new perspective of regularizing distortion-minimizing feedback communication through information maximization, which enables us to find the optimal solution that otherwise would be intractable.