Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework

2025-05-27

AI Summary

Representation learning for structured data—such as images, point clouds, graphs, meshes, and manifolds—in deep learning faces fundamental challenges: modeling long-range dependencies, alleviating oversmoothing in graph neural networks (GNNs), handling heterophily, and adapting to non-Euclidean geometries. Method: This paper introduces the first topological neural network framework grounded in *copresheaves*, leveraging algebraic topology and category theory to unify representation learning across diverse structured domains. It systematically incorporates copresheaf theory into deep learning—yielding a categorical unification of CNNs, GNNs, MeshNets, and beyond—and constructs a provably sound, multi-scale architectural design space that balances theoretical rigor with practical efficiency. Contribution/Results: The framework achieves state-of-the-art performance on multiple benchmarks, significantly outperforming conventional baselines—particularly on hierarchical modeling and local-sensitive tasks—while providing principled, topology-aware generalization across data modalities.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Representation Learning for Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We introduce copresheaf topological neural networks (CTNNs), a powerful and 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 grounding model design in the language of copresheaves, a concept from algebraic topology that generalizes and subsumes 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: 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 underscore CTNNs as a principled, multi-scale foundation for the next generation of deep learning architectures.

Problem

• Generalizing deep learning for diverse structured data types
• Addressing challenges in representation learning and model design
• Improving performance on hierarchical and localized sensitivity tasks

Innovation

• Generalized framework using copresheaves from topology
• Unifies diverse deep learning architectures for structured data
• Addresses challenges like long-range dependencies and non-Euclidean domains