3D Weighted Geometric Graph Neural Networks for Sheep Facial Pain Assessment

Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule. These second-order representations are naturally captured by points on the symmetric positive definite matrices (SPD) manifold; (2) Standard message passing applies shared transformations across edges. Sheaf neural networks address this via edge-specific transformations, but existing formulations remain confined to vector spaces and therefore cannot propagate matrix-valued features. We address both challenges by developing the first sheaf neural network operates natively on the SPD manifold. Our key insight is that the SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space. Theoretically, we prove that SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations. Empirically, our sheaf convolution transforms effectively rank-1 directional inputs into full-rank matrices encoding local geometric structure. Our dual-stream architecture achieves SOTA on 6/7 MoleculeNet benchmarks, with the sheaf framework providing consistent depth robustness.

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.

Automatic pain assessment from facial video remains challenging due to the spatial heterogeneity of pain-related facial cues. This study proposes ReFace, a spatial reorganization pipeline that divides facial input into four spatial quadrants before tokenization, rather than processing the entire face as a single region. Evaluated on the AI4Pain dataset, the proposed approach achieves $56.00\%$ accuracy on the test set using video only, achieving the highest reported accuracy under the fixed AI4Pain benchmark protocol among the compared methods. Notably, the four-quadrant configuration processes the same total pixel budget as the full-face input, yet achieves higher accuracy, suggesting that spatial reorganization can improve performance under the proposed tokenization design. A single quadrant region, processing just one quarter of those pixels, remains competitive at a fraction of the computational cost.