Contextuality, Holonomy and Discrete Fiber Bundles in Group-Valued Boltzmann Machines
This work proposes a geometric extension for AI architectures to handle relational data, but it appears incremental as it builds on existing RBMs with group-valued weights.
The authors tackled the problem of modeling complex relational structures in restricted Boltzmann machines by generalizing weights to abstract groups, enabling applications in vision, language, and quantum learning. They introduced a contextuality index based on group-valued holonomies to quantify global inconsistencies, linking it to sheaf theory and gauge theory.
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.