Sheaf theory: from deep geometry to deep learning

2025-02-21

AI Summary

Deep learning and data science lack principled frameworks for modeling the geometric structure of non-Euclidean, heterogeneous data. Method: This work establishes, for the first time, a systematic mapping between sheaf theory and deep learning. We propose a fast cohomology computation algorithm applicable to arbitrary finite posets, integrating sheaf Laplacians, sheaf diffusion, and derived functors to construct a computationally tractable sheaf-theoretic machine learning pipeline—from mathematical definition to neural network module. Contributions: (1) A unified geometric representation framework for non-Euclidean data; (2) Identification of fundamental theoretical blind spots in existing ML models regarding structural awareness; (3) Formulation of several open problems bridging category theory with signal processing and deep learning—each combining theoretical depth with practical applicability—to catalyze interdisciplinary advancement.

Technology Category

• Machine Learning: Deep Learning Theory
• Computer Vision: Other Foundations of Computer 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: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Machine learning and data science for the Web

Abstract

This paper provides an overview of the applications of sheaf theory in deep learning, data science, and computer science in general. The primary text of this work serves as a friendly introduction to applied and computational sheaf theory accessible to those with modest mathematical familiarity. We describe intuitions and motivations underlying sheaf theory shared by both theoretical researchers and practitioners, bridging classical mathematical theory and its more recent implementations within signal processing and deep learning. We observe that most notions commonly considered specific to cellular sheaves translate to sheaves on arbitrary posets, providing an interesting avenue for further generalization of these methods in applications, and we present a new algorithm to compute sheaf cohomology on arbitrary finite posets in response. By integrating classical theory with recent applications, this work reveals certain blind spots in current machine learning practices. We conclude with a list of problems related to sheaf-theoretic applications that we find mathematically insightful and practically instructive to solve. To ensure the exposition of sheaf theory is self-contained, a rigorous mathematical introduction is provided in appendices which moves from an introduction of diagrams and sheaves to the definition of derived functors, higher order cohomology, sheaf Laplacians, sheaf diffusion, and interconnections of these subjects therein.

Problem

• Bridging sheaf theory and deep learning
• Generalizing sheaf methods to arbitrary posets
• Addressing blind spots in machine learning

Innovation

• Sheaf theory in deep learning
• Sheaf cohomology on posets
• Bridging classical and modern applications