[Paper Review] Information Structures of Capacity Achieving Distributions for Feedback Channels with Memory and Transmission Cost: Stochastic Optimal Control & Variational Equalities-Part I.

Christos K. Kourtellaris, Charalambos D. Charalambous|arXiv (Cornell University)|Dec 14, 2015

Wireless Communication Security Techniques29 references20 citations

TL;DR

This paper characterizes the finite-time feedback information (FTFI) capacity for memory-structured channels with transmission cost constraints using stochastic optimal control and variational equalities. It proves that optimal input distributions depend only on the past M output symbols, reducing the information structure to a finite-memory form and enabling exact capacity computation via directed information maximization over this subset.

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, $B^n = \{B_j: j=0,1, \ldots, n\}$ are the channel outputs and $A^n=\{A_j: j=0,1, \ldots, n\}$ are the channel inputs. The characterizations of FTFI capacity, are obtained by first identifying the information structures of the optimal channel input conditional distributions ${\cal P}_{[0,n]} =\big\{ {\bf P}_{A_i|A^{i-1}, B^{i-1}}: i=1, \ldots, n\big\}$, which maximize directed information $C_{A^n ightarrow B^n}^{FB} = \sup_{ {\cal P}_{[0,n]} } I(A^n ightarrow B^n), I(A^n ightarrow B^n) = \sum_{i=0}^n I(A^i;B_i|B^{i-1}) . $ The main theorem states, for any channel with memory $M$, the optimal channel input conditional distributions occur in the subset ${\cal Q}_{[0,n]}= \big\{ {\bf P}_{A_i|B_{i-M}^{i-1}}: i=1, \ldots, n\big\}$, and the characterization of FTFI capacity is given by $C_{A^n ightarrow B^n}^{FB, M} = \sup_{ {\cal Q}_{[0,n]} } \sum_{i=0}^n I(A_i; B_i|B_{i-M}^{i-1})$ . Similar conclusions are derived for problems with transmission cost constraints. The methodology utilizes stochastic optimal control theory, to identify the control process, the controlled process, and a variational equality of directed information, to derive upper bounds on $I(A^n ightarrow B^n)$, which are achievable over specific subsets of channel input conditional distributions. For channels with limited memory, this implies the transition probabilities of the channel output process are also of limited memory.

Motivation & Objective

• To characterize the finite-time feedback information (FTFI) capacity for discrete-time memory channels with feedback.
• To identify the optimal information structure of channel input distributions that maximize directed information under feedback and transmission cost constraints.
• To establish that for channels with memory M, the optimal input distribution depends only on the last M output symbols, not the entire history.
• To derive a closed-form expression for FTFI capacity using a variational equality and stochastic control framework.

Proposed method

• Formulates the feedback capacity problem as a stochastic optimal control problem, identifying the control process (channel input) and controlled process (channel output).
• Derives a variational equality for directed information to establish upper bounds on $ I(A^n \rightarrow B^n) $, which are tight under specific input distributions.
• Identifies the optimal information structure as $ \mathcal{Q}_{[0,n]} = \{ \mathbf{P}_{A_i|B_{i-M}^{i-1}} \} $, showing that dependence on past inputs is unnecessary.
• Uses this structure to re-express the directed information as $ \sum_{i=0}^n I(A_i; B_i|B_{i-M}^{i-1}) $, enabling capacity computation over a finite-memory subset.
• Applies the framework to both memoryless and memory-structured channels, proving achievability of the upper bound under the derived input distribution class.
• Establishes that the channel output process also inherits limited memory, with transition probabilities depending on only the last M outputs.

Experimental results

Research questions

• RQ1 What is the optimal information structure for channel input distributions that maximize directed information in feedback channels with memory M and transmission cost constraints?
• RQ2 Can the feedback capacity of such channels be characterized using a finite-memory information structure rather than full history dependence?
• RQ3 How does stochastic optimal control theory enable the derivation of variational equalities for directed information in feedback settings?
• RQ4 What is the precise relationship between the memory of the channel and the memory of the optimal input distribution?
• RQ5 Is the upper bound on directed information achievable, and if so, over which subset of input distributions?

Key findings

• The optimal channel input conditional distributions for feedback channels with memory M belong to the subset $ \mathcal{Q}_{[0,n]} = \{ \mathbf{P}_{A_i|B_{i-M}^{i-1}} \} $, which depends only on the last M output symbols.
• The FTFI capacity for channels with memory M is given by $ C_{A^n \rightarrow B^n}^{FB, M} = \sup_{\mathcal{Q}_{[0,n]}} \sum_{i=0}^n I(A_i; B_i|B_{i-M}^{i-1}) $, providing a computable expression for capacity.
• The directed information $ I(A^n \rightarrow B^n) $ is bounded above by this expression, and the bound is achievable using the derived input distribution class.
• The channel output process $ B^n $ inherits a finite-memory structure, with transition probabilities depending only on the last M outputs.
• The methodology using stochastic optimal control and variational equalities enables the derivation of tight upper bounds on directed information that are achievable under the specified information structure.
• For channels with transmission cost constraints, similar characterizations hold, showing that the optimal input distribution remains dependent only on the last M outputs, preserving the finite-memory structure.