[论文] Attention as Lattice Gauge Theory: Relations Interact Only Through Wil...

论文概要

研究领域: ML理论 / 格点规范 作者: Aakash Sarkar, Ananya Mukherjee, Motoshige Sato 发布时间: 2026-10-27 arXiv: 2610.37656

中文摘要

关系通过共享实体组合——Alice是Bob的母亲，Bob是Carol的父亲，所以Alice是Carol的祖母——且顺序至关重要：父亲的母亲不等于母亲的父亲。验证组合关系意味着闭合一个回路。本文证明，如果每个实体的坐标系可以独立旋转（逐实体框架自由度），则三个结论成立：在纯规范场中，不共享环路的关系完全无法交互；三角形唯一的规范不变得分是其全纯量H = T_ij·T_jk·T_ki的类函数——即Wilson loop；边的唯一等变更新形式为U = Ψ(H)·T_ij。因此，基于Wilson loop的边注意力不是众多设计之一，而是对称性允许的唯一形式。三个社区各自持有该形式的一部分：知识图谱的交换算子嵌入是其阿贝尔（交换）部分；AlphaFold的三角形更新和Edge Transformer有其形状但缺少约束；格点QCD在格点上等变地构建了它。本文在一般关系图上实例化等变子族为全纯注意力（holonomy attention），无需训练即可在20%边被腐蚀的稀疏图上以0.88的准确率推断未观测的S₄关系。

原文摘要

Relations compose through the entity they share - Alice is Bob's mother and Bob is Carol's father, so Alice is Carol's grandmother - and the order matters, since a father's mother is not a mother's father; verifying a composed relation means closing the loop. We show that if each entity's coordinate frame may be rotated independently (per-entity frame freedom), three things follow. In the pure-gauge sector, relations that share no cycle cannot interact at all. The only gauge-invariant score of a triangle is a class function of its holonomy H = T_{ij} T_{jk} T_{ki} - a Wilson loop. And the only equivariant update of an edge is U = Ψ(H)\, T_{ij} with Ψ conjugation-equivariant. Edge attention by Wilson loops is therefore not one design among many but the unique form the symmetry permits. Three communities hold pieces of this form: the commuting-operator embeddings of knowledge graphs are its abelian (commuting) sector; AlphaFold's triangle update and the Edge Transformer have its shape - an edge updated through the third vertex - without its constraints; and lattice QCD builds it equivariantly on lattices. We instantiate the equivariant sub-family on general relational graphs as holonomy attention. With no training and composition alone, it infers unobserved S_4 relations on sparse graphs at 0.88 with a fifth of the edges corrupted, at or slightly below specialist synchronization algorithms, where every commutative architecture is capped at 0.083; its holonomy score flags corrupted triangles at AUC 0.99 where commuting scores are capped at 0.76 by a second selection rule, and it separates clusters without labels.

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*自动采集于 2026-10-29*

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