Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework
This work addresses a foundational open problem in deep learning by providing a theoretical framework for designing architectures tailored to diverse data types, which could impact fields like computer vision and graph analysis.
The paper tackles the challenge of principled neural architecture design for structured data by introducing copresheaf topological neural networks (CTNNs), a unifying framework that generalizes existing models and demonstrates consistent performance improvements over conventional baselines on benchmarks.
We introduce copresheaf topological neural networks (CTNNs), a powerful unifying framework that encapsulates a wide spectrum of deep learning architectures, designed to operate on structured data, including images, point clouds, graphs, meshes, and topological manifolds. While deep learning has profoundly impacted domains ranging from digital assistants to autonomous systems, the principled design of neural architectures tailored to specific tasks and data types remains one of the field's most persistent open challenges. CTNNs address this gap by formulating model design in the language of copresheaves, a concept from algebraic topology that generalizes most practical deep learning models in use today. This abstract yet constructive formulation yields a rich design space from which theoretically sound and practically effective solutions can be derived to tackle core challenges in representation learning, such as long-range dependencies, oversmoothing, heterophily, and non-Euclidean domains. Our empirical results on structured data benchmarks demonstrate that CTNNs consistently outperform conventional baselines, particularly in tasks requiring hierarchical or localized sensitivity. These results establish CTNNs as a principled multi-scale foundation for the next generation of deep learning architectures.