Sheaf theory: from deep geometry to deep learning (2025)

A new paper provides an accessible overview of sheaf theory applications in deep learning, data science, and computer science, bridging classical mathematics with modern implementations in signal processing and deep learning. The authors present a novel algorithm for computing sheaf cohomology on arbitrary finite posets and identify blind spots in current machine learning practices.

Mathematics Algebraic Topology Submitted on 21 Feb 2025 Title:Sheaf theory: from deep geometry to deep learning View PDF /pdf/2502.15476 HTML experimental https://arxiv.org/html/2502.15476v1 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. Current browse context: math.AT References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .