Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework

TL;DR

This paper introduces copresheaf topological neural networks (CTNNs), a unifying framework based on algebraic topology that generalizes many deep learning architectures for structured data, improving performance on various benchmarks.

Contribution

The paper presents CTNNs, a novel topological framework that unifies and extends deep learning architectures for structured data, addressing core challenges with a principled, mathematically grounded approach.

Findings

CTNNs outperform traditional models on structured data benchmarks.

They effectively handle hierarchical and localized data sensitivities.

The framework offers a versatile foundation for future deep learning architectures.

Abstract

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.