🤖 AI Summary
This work investigates whether the predictions of geometric neural networks—such as sheaf neural networks—genuinely rely on intrinsic geometric mechanisms like SO(2) rotations and parallel transport, rather than merely reflecting task performance indirectly. To this end, the authors propose the first basis-invariant triangular loop product metric and employ intervention experiments to disentangle the effects of rotation, stalk-space area, and orientation. Evaluation on a custom high-homophily GraphUniverse dataset demonstrates that neural sheaf propagation (NSP) substantially enhances loop rotation (up to 0.388 radians), while substituting identity transport drastically increases error, confirming the critical role of geometric mechanisms in connection sensitivity. Furthermore, a graph-summary ridge regression predictor outperforms others, underscoring the importance of structural inductive bias.
📝 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.