Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning

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.

3D Medical Imaging Segmentation on Non-Contrast CT

This technical report analyzes non-contrast CT image segmentation in computer vision. It revisits a proposed method, examines the background of non-contrast CT imaging, and highlights the significance of segmentation. The study reviews representative methods, including convolutional-based and CNN-Transformer hybrid approaches, discussing their contributions, advantages, and limitations. The nnUNet stands out as the state-of-the-art method across various segmentation tasks. The report explores the relationship between the proposed method and existing approaches, emphasizing the role of global context modeling in semantic labeling and mask generation. Future directions include addressing the long-tail problem, utilizing pre-trained models for medical imaging, and exploring self-supervised or contrastive pre-training techniques. This report offers insights into non-contrast CT image segmentation and potential advancements in the field.