Poset-Markov Channels: Capacity via Group Symmetry

TL;DR

This paper introduces poset-causal channels, a new formalism for channels with memory indexed by posets, and develops a symmetry-based method to bound their feedback capacity, demonstrated on specific models.

Contribution

It presents a novel formalism for channels with poset-structured memory and a symmetry reduction technique to derive capacity bounds for a subclass of these channels.

Findings

Derived a single-letter upper bound on feedback capacity for poset-causal channels with Markov symmetry.

Bound is tight for the NOST channel model.

Extended the NOST model to a two-dimensional version.

Abstract

Computing channel capacity is in general intractable because it is given by the limit of a sequence of optimization problems whose dimensionality grows to infinity. As a result, constant-sized characterizations of feedback or non-feedback capacity are known for only a few classes of channels with memory. This paper introduces poset-causal channels$\unicode{x2014}$a new formalism of a communication channel in which channel inputs and outputs are indexed by the elements of a partially ordered set (poset). We develop a novel methodology that allows us to establish a single-letter upper bound on the feedback capacity of a subclass of poset-causal channels whose memory structure exhibits a Markov property and symmetry. The methodology is based on symmetry reduction in optimization. We instantiate our method on two channel models: the Noisy Output is The STate (NOST) channel$\unicode{x2014}$for which the bound is tight$\unicode{x2014}$and a new two-dimensional extension of it.