探究图神经网络中的几何机制是否真实影响预测结果
Do Sheaf Neural Networks Use Holonomy? A Measure--Intervene--Control Study
- 通过干预边传输与环路读出，检验拓扑连接的演化
- 三角计数任务使旋转角度从0.010增至0.388弧度，社区检测仅0.029
- 虽能学习非平凡连接，但未证明旋转变换驱动预测
几何架构常以内在机制为动机，但精度本身无法验证其是否被实际使用。在切层神经网络（SNNs）中，边传输构成连接，其环路乘积定义了全同性。我们研究训练是否改变三角形全同性、预测是否依赖学习到的连接，以及全同性是否驱动三角形计数。采用基无关环路读出、恒等干预和捷径控制。在高同质性GraphUniverse图上，三角形计数使神经切层传播（NSP）中的平均SO(2)三角旋转从0.010提升至0.388弧度，而社区检测仅达0.029弧度。数据更多时，学习型SO(2)-NSP优于恒等NSP，且训练后替换传输进一步增加误差。然而，岭回归更准确，对角映射无需连续旋转即可改进，固定度模型在无计数提升下仍发展旋转。因此，NSP可学习并依赖非平凡连接，但实验未显示三角形全同性驱动其预测。
原文摘要 · Abstract (English)
Geometric architectures are often motivated by internal mechanisms, but accuracy alone does not show whether predictions use them. In Sheaf Neural Networks (SNNs), edge transports form a connection whose cycle products define holonomy. We ask whether training changes triangle holonomy, whether predictions rely on the learned connection, and whether holonomy drives triangle counting. We use basis-independent loop readouts with identity interventions and shortcut controls. On high-homophily GraphUniverse graphs, triangle counting increases the mean SO(2) triangle rotation in Neural Sheaf Propagation (NSP) from 0.010 to 0.388 radians, while community detection ends at 0.029 radians. With more data, learned SO(2)--NSP outperforms Identity NSP, and replacing its transports after training increases error further. However, ridge regression is more accurate, diagonal maps improve without continuous rotation, and fixed-degree models develop rotation without improved counting. Thus, NSP can learn and rely on a nontrivial connection, but our experiments do not show that triangle holonomy drives its predictions.
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