# Contextuality, Holonomy and Discrete Fiber Bundles in Group-Valued Boltzmann Machines

arXiv: https://arxiv.org/abs/2509.10536 · HTML: https://mlsanity.com/paper/2509.10536/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 33 references that are in the map.
- First posted on arXiv 5 Sep 2025.

## Builds on (most cited first, 3 of 3)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Contextuality: At the Borders of Paradox](https://mlsanity.com/paper/2011.04899/) (2020)

## Abstract (arXiv)

We propose a geometric extension of restricted Boltzmann machines (RBMs) by allowing weights to take values in abstract groups such as ( GLn(ℝ) ), ( 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 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.