| Abstract: |
| Combinatorial structures such as graphs, simplicial complexes, and cell complexes provide the foundations for geometric and topological deep learning (GDL and TDL), yet the mechanisms governing how features propagate and interact across local and global structures during training remain not yet fully understood. In this talk, we present an ongoing project that develops a cellular sheaf-theoretic framework to study the consistency and interaction of learned signals on such mathematical objects. By encoding node features and edge weights as local sections of a cellular sheaf, the proposed approach enables a systematic analysis of local agreements and consistencies, providing a new perspective on feature diffusion and aggregation from a topological viewpoint. In addition, a multiscale extension inspired by topological data analysis is introduced to capture hierarchical feature interactions across different resolutions. This framework aims to provide a unified mathematical perspective on GDL/TDL models, with potential applications to tasks such as node classification, substructure detection, and community detection. This is a work in progress, and preliminary results have been presented at the AAAI 2026 Workshop on Foretell of Future AI from Mathematical Foundation. |
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