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

TL;DR

This paper introduces a categorical framework for belief propagation that unifies various inference methods and detects obstructions using sheaf theory, enabling exact inference and efficiency improvements.

Contribution

It develops a sheaf-theoretic foundation for belief propagation, introduces the HATCC algorithm for detecting obstructions, and demonstrates practical speedups and exact inference.

Findings

Exact inference achieved with speedup over junction trees.

HATCC detects holonomy obstructions in factor graphs.

Effective for grid MRFs, random graphs, and SAT instances.

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.