Learned, Relied Upon, or Necessary? Separating Checkpoint Dependence from Task-Level Value in Sheaf GNNs

Sheaf Graph Neural Networks (GNNs) learn restriction maps on node-edge incidences to transport features into comparable coordinates. This is often interpreted as the model discovering useful edge geometry. This paper investigates whether this transport is truly necessary for task performance or if it is merely a byproduct of the optimization process that could be replaced by simpler, non-geometric alternatives.

The author introduces a rigorous audit framework to separate three distinct claims: (1) map movement (the maps changed during training), (2) checkpoint reliance (the current model fails if maps are replaced), and (3) protocol-relative replacement (a retrained model without the original map structure can match the performance). The study uses a task-null theorem to show that labels only constrain specific classifier directions, leaving many degrees of freedom in the maps invisible to the task. The author then audits public implementations of NSD and DNSD across five benchmarks, comparing them against controls like identity maps, shared-map families, and resampled-assignment models.

The audit reveals that while all tested checkpoints rely on their learned maps (checkpoint reliance), this does not imply the maps are necessary. On four out of five benchmarks, retraining a model with assignment-breaking controls or shared-map architectures recovers the performance of the full model. Only the Roman-Empire dataset shows a persistent, statistically significant advantage for the original learned transport. The author concludes that learned maps can govern a fitted computation without constituting indispensable edge geometry, and suggests that claims of learned transport should always be paired with matched retraining experiments.

This work provides a necessary methodological correction for the field of geometric deep learning. It demonstrates that post-hoc ablation studies—which are common in the literature—are insufficient to prove that a model has learned meaningful geometric structure. By distinguishing between reliance and necessity, the paper encourages researchers to use more robust controls to verify whether their models are truly exploiting graph geometry or simply fitting to the data through high-capacity, non-essential parameters.