Information Structures for Feedback Capacity of Channels with Memory and Transmission Cost: Stochastic Optimal Control & Variational Equalities-Part I

TL;DR

This paper characterizes the feedback capacity of channels with memory and transmission costs using stochastic control and variational methods, identifying optimal input structures and deriving capacity formulas.

Contribution

It introduces a novel information structure characterization for optimal channel inputs in channels with memory, extending Shannon's capacity concepts to more complex channels.

Findings

Optimal input distributions satisfy conditional independence with respect to channel memory

Capacity characterized by supremum of conditional mutual information terms

Methodology applies stochastic control and variational equalities to derive bounds

Abstract

The Finite Transmission Feedback Information (FTFI) capacity is characterized for any class of channel conditional distributions $\big\{{\bf P}_{B_i|B^{i-1}, A_i} :i=0, 1, \ldots, n\big\}$ and $\big\{ {\bf P}_{B_i|B_{i-M}^{i-1}, A_i} :i=0, 1, \ldots, n\big\}$, where $M$ is the memory of the channel, $B^n {\stackrel{\triangle}{=}} \{B_j: j=\ldots, 0,1, \ldots, n\}$ are the channel outputs and $A^n{\stackrel{\triangle}{=}} \{A_j: j=\ldots, 0,1, \ldots, n\}$ are the channel inputs. The characterizations of FTFI capacity, are obtained by first identifying the information structures of the optimal channel input conditional distributions ${\cal P}_{[0,n]} {\stackrel{\triangle}{=}} \big\{ {\bf P}_{A_i|A^{i-1}, B^{i-1}}: i=0, \ldots, n\big\}$, which maximize directed information. The main theorem states, for any channel with memory $M$, the optimal channel input conditional distributions occur in the subset satisfying conditional independence $\stackrel{\circ}{\cal P}_{[0,n]}{\stackrel{\triangle}{=}} \big\{ {\bf P}_{A_i|A^{i-1}, B^{i-1}}= {\bf P}_{A_i|B_{i-M}^{i-1}}: i=1, \ldots, n\big\}$, and the characterization of FTFI capacity is given by $C_{A^n \rightarrow B^n}^{FB, M} {\stackrel{\triangle}{=}} \sup_{ \stackrel{\circ}{\cal P}_{[0,n]} } \sum_{i=0}^n I(A_i; B_i|B_{i-M}^{i-1}) $. The methodology utilizes stochastic optimal control theory and a variational equality of directed information, to derive upper bounds on $I(A^n \rightarrow B^n)$, which are achievable over specific subsets of channel input conditional distributions ${\cal P}_{[0,n]}$, which are characterized by conditional independence. For any of the above classes of channel distributions and transmission cost functions, a direct analogy, in terms of conditional independence, of the characterizations of FTFI capacity and Shannon's capacity formulae of Memoryless Channels is identified.