将玻尔兹曼机权重扩展到抽象群，用几何曲率建模复杂关系。
Contextuality, Holonomy and Discrete Fiber Bundles in Group-Valued Boltzmann Machines
- 权重取值于群（如SU(2)、GL_n(R)），支持投影变换与自旋动力学建模
- 提出基于群值全息的上下文指数，量化局部权重导致的全局不一致性
- 适用于视觉、语言及量子学习，适合研究拓扑与几何约束的模型
我们提出一种几何扩展的受限玻尔兹曼机（RBMs），允许权重取值于抽象群，如 $\mathrm{GL}_n(\mathbb{R})$、$\mathrm{SU}(2)$，甚至无限维算子群。该推广使模型能够刻画复杂的相对结构，包括射影变换、旋量动力学和函数对称性，在视觉、语言和量子学习中具有直接应用。本文核心贡献是引入一个基于群值全息的上下文指数，通过计算RBMs图中环路的全息来量化局部权重引起的全局不一致或“曲率”，推广了经典的一致性、相干性与几何平坦性概念。建立了与层化上下文性、规范理论和非交换几何的联系，并提供了有限与无限维的数值与图示例。该框架为人工智能开辟了新方向，包括曲率感知学习架构和不确定或对抗环境下的拓扑正则化。
原文摘要 · Abstract (English)
We propose a geometric extension of restricted Boltzmann machines (RBMs) by allowing weights to take values in abstract groups such as \( \mathrm{GL}_n(\mathbb{R}) \), \( \mathrm{SU}(2) \), or even infinite-dimensional operator groups. This generalization enables the modeling of complex relational structures, including projective transformations, spinor dynamics, and functional symmetries, with direct applications to vision, language, and quantum learning. A central contribution of this work is the introduction of a \emph{contextuality index} based on group-valued holonomies computed along cycles in the RBM graph. This index quantifies the global inconsistency or "curvature" induced by local weights, generalizing classical notions of coherence, consistency, and geometric flatness. We establish links with sheaf-theoretic contextuality, gauge theory, and noncommutative geometry, and provide numerical and diagrammatic examples in both finite and infinite dimensions. This framework opens novel directions in AI, from curvature-aware learning architectures to topological regularization in uncertain or adversarial environments.
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