Do Sheaf Neural Networks Use Holonomy? A Measure--Intervene--Control Study

Title:Do Sheaf Neural Networks Use Holonomy? A Measure--Intervene--Control Study

Abstract:Geometric architectures are often justified by internal mechanisms such as rotations, yet task performance alone cannot show whether those mechanisms drive predictions. Using sheaf neural networks (SNNs) as a testbed, we introduce the first basis-independent measurement of trained triangle-loop products, separating rotation, stalk-space area, and orientation. In a custom high-homophily GraphUniverse regime, Neural Sheaf Propagation (NSP) increases the triangle-weighted mean two-dimensional SO(2) loop rotation from 0.010 to 0.388 radians for triangle counting, while the community-detection comparison ends at 0.029 radians. Across the training-set-size experiment, replacing all learned SO(2) transports with identities sharply increases test error, establishing post-training sensitivity to the complete learned connection. However, a graph-summary ridge predictor is more accurate, diagonal maps also improve, and fixed-degree graphs develop increasing rotation without outperforming the training-mean predictor. This measure-intervene-control study separates geometric change, connection sensitivity, and evidence for triangle-specific computation.

Machine Learning (cs.LG)

arXiv:2607.19514 [cs.LG]

(or arXiv:2607.19514v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2607.19514 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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