Poset-Markov Channels: Capacity via Group Symmetry

2025-06-24

AI Summary

Computing the capacity of channels with memory typically suffers from the curse of dimensionality, with single-letter characterizations known only for a few special cases. To address this, we introduce the “partially ordered set (poset) causal channel” model, where inputs and outputs are indexed by a poset to uniformly represent diverse memory structures. For the subclass exhibiting Markovian dynamics and group symmetry, we integrate poset-based causal modeling, symmetry-driven state-space reduction, and optimization-based dimensionality reduction to derive, for the first time, a tight single-letter upper bound on the feedback capacity. Our approach transcends conventional frameworks restricted to linear time or finite-state assumptions: it yields a tight capacity bound for the NOST channel and extends successfully to a two-dimensional spatially extended variant. This work establishes a new analytical and generalizable paradigm for capacity characterization of complex memory channels.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Computing channel capacity is intractable for most memory channels
• Introduces poset-causal channels with inputs/outputs indexed by posets
• Develops single-letter upper bound for feedback capacity using symmetry

Innovation

• Poset-causal channels with Markov property
• Symmetry reduction in optimization
• Single-letter feedback capacity bound