基于范畴的信念传播：通过降阶和holonomy的sheaf理论推断
Categorical Belief Propagation: Sheaf-Theoretic Inference via Descent and Holonomy
AI总结：
本文提出HATCC算法，通过sheaf理论框架统一精确推断与循环BP失败，利用holonomy计算检测降阶障碍并提升效率。
摘要：
我们为因子图上的信念传播建立了范畴基础。我们构建了在带类型签名上自由的超图范畴$\Syn_Σ$，并证明了其普遍属性，从而通过唯一到矩阵范畴$\cat{Mat}_R$的函子获得组合语义。消息传递是通过一个Grothendieck纤维$\int\Msg \to \cat{FG}_Σ$在极化因子图上进行的，用schedule索引的内射定义BP更新。我们刻画精确推断为有效的降阶：当重叠上的兼容条件成立时，局部信念形成降阶数据。这一框架将树精确性、junction树算法和循环BP失败统一在sheaf理论障碍之下。我们引入HATCC（Holonomy-aware Tree Compilation），一种算法，通过在因子神经上计算holonomy来检测降阶障碍，将非平凡holonomy编译为模式变量，并在增强图上降级为树BP。复杂度为$O(n^2 d_{\max} + c \cdot k_{\max} \cdot δ_{\max}^3 + n \cdot δ_{\max}^2)$，其中$n$为因素数，$c$为基本循环数。实验结果展示了在网格MRF和随机图上的精确推断，比junction树有显著加速，同时在可满足性实例上检测到UNSAT。
英文摘要：
We develop a categorical foundation for belief propagation on factor graphs. We construct the free hypergraph category \(\Syn_Σ\) 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 \(\int\Msg \to \cat{FG}_Σ\) 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 \(O(n^2 d_{\max} + c \cdot k_{\max} \cdot δ_{\max}^3 + n \cdot δ_{\max}^2)\) 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
