A Categorical Treatment of Open Linear Systems

2024-03-06

arXiv.org

AI Summary

Formalizing open stochastic systems under combined uncertainty from incomplete information and probabilistic perturbations remains challenging within categorical frameworks. Method: We introduce *copartiality* to model uncertainty propagation, categorify Willems-style open stochastic systems for the first time, define *extended Gaussian distributions*—a unification of Gaussian probability measures and linear relations—and construct the *category of extended Gaussian maps* to support compositional modeling and semantic consistency verification. Contribution: This work bridges categorical probability theory and control theory (e.g., signal-flow graphs), rigorously formalizes physical noise constraints and Bayesian noninformative priors, and establishes a novel structural paradigm for modeling and analyzing open linear stochastic systems. It provides a unified categorical foundation wherein stochasticity, partiality, and linearity are coherently integrated, enabling principled compositionality, abstraction, and verification in uncertain dynamical systems.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

An open stochastic system `a la Jan Willems is a system affected by two qualitatively different kinds of uncertainty: one is probabilistic fluctuation, and the other one is nondeterminism caused by a fundamental lack of information. We present a formalization of open stochastic systems in the language of category theory. Central to this is the notion of copartiality which models how the lack of information propagates through a system (corresponding to the coarseness of sigma-algebras in Willems' work). As a concrete example, we study extended Gaussian distributions, which combine Gaussian probability with nondeterminism and correspond precisely to Willems' notion of Gaussian linear systems. We describe them both as measure-theoretic and abstract categorical entities, which enables us to rigorously describe a variety of phenomena like noisy physical laws and uninformative priors in Bayesian statistics. The category of extended Gaussian maps can be seen as a mutual generalization of Gaussian probability and linear relations, which connects the literature on categorical probability with ideas from control theory like signal-flow diagrams.

Problem

• Modeling open stochastic systems with probabilistic and nondeterministic uncertainty
• Formalizing copartiality to track information lack propagation
• Connecting categorical probability with control theory concepts

Innovation

• Formalizes open stochastic systems categorically
• Introduces copartiality for information lack propagation
• Models extended Gaussian distributions abstractly