Categorical Belief Propagation: Sheaf-Theoretic Inference via Descent and Holonomy

2026-01-08

arXiv.org

AI Summary

This work addresses the failure of conventional belief propagation in factor graphs containing loopy structures by reconstructing its mathematical foundations through category theory and sheaf theory. It introduces, for the first time, the notions of effective descent conditions and sheaf-theoretic obstructions to characterize exact inference. The proposed HATCC algorithm models message passing as schedule-indexed endomorphisms over a Grothendieck fibration, integrating tools from hypergraph categories, matrix category functors, factor neural complexes, and holonomy computations to detect and handle nontrivial topological obstructions. Empirical results demonstrate that this approach achieves exact inference more efficiently than junction tree methods on both grid Markov random fields and random graphs, and it successfully applies to UNSAT detection in SAT problems.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We develop a categorical foundation for belief propagation on factor graphs. We construct the free hypergraph category \(\Syn_\Sigma\) on a typed signature and prove its universal property, yielding compositional semantics via a unique functor to the matrix category \(\cat{Mat}_R\). Message-passing is formulated using a Grothendieck fibration \(\int\Msg \to \cat{FG}_\Sigma\) over polarized factor graphs, with schedule-indexed endomorphisms defining BP updates. We characterize exact inference as effective descent: local beliefs form a descent datum when compatibility conditions hold on overlaps. This framework unifies tree exactness, junction tree algorithms, and loopy BP failures under sheaf-theoretic obstructions. We introduce HATCC (Holonomy-Aware Tree Compilation), an algorithm that detects descent obstructions via holonomy computation on the factor nerve, compiles non-trivial holonomy into mode variables, and reduces to tree BP on an augmented graph. Complexity is \(O(n^2 d_{\max} + c \cdot k_{\max} \cdot \delta_{\max}^3 + n \cdot \delta_{\max}^2)\) for \(n\) factors and \(c\) fundamental cycles. Experimental results demonstrate exact inference with significant speedup over junction trees on grid MRFs and random graphs, along with UNSAT detection on satisfiability instances.

Problem

• belief propagation
• factor graphs
• exact inference
• sheaf theory
• holonomy

Innovation

• Categorical Belief Propagation
• Sheaf Theory
• Effective Descent
• Holonomy
• HATCC