Sheaf theory: from deep geometry to deep learning

TL;DR

An overview paper introducing applications of sheaf theory in deep learning and data science, presenting a new algorithm to compute sheaf cohomology on arbitrary finite posets.

Problem

Sheaf theory is a powerful tool from topology and geometry, but its mathematical barriers make it difficult for deep learning researchers to access. Moreover, existing computational methods for sheaf theory are mostly limited to cellular sheaves, necessitating generalization to arbitrary posets.

Approach

This paper explains the intuition and motivation behind sheaf theory in a way accessible to researchers with modest mathematical background. It bridges classical mathematical theory with recent implementations in signal processing and deep learning, showing that notions specific to cellular sheaves can be generalized to arbitrary finite posets. Accordingly, it proposes a new algorithm to compute sheaf cohomology on arbitrary finite posets.

Results & Contribution

This work reveals blind spots in current machine learning practices regarding sheaf-theoretic applications and presents a list of problems that are mathematically insightful and practically instructive to solve. The appendices provide a rigorous mathematical introduction to sheaf theory, enabling readers to understand derived functors, higher-order cohomology, sheaf Laplacians, sheaf diffusion, and their interconnections.