Category-Theoretical and Topos-Theoretical Frameworks in Machine Learning: A Survey

2024-08-26

arXiv.org

AI Summary

This paper addresses the lack of a unified mathematical foundation for machine learning. Methodologically, it introduces the first topos-theoretic machine learning framework, systematically integrating functorial composition modeling, sheaf theory, higher-order type theory, and topos semantics to unify gradient-based, probabilistic, invariance/equivalence-based, and topological learning paradigms. Its core contributions are threefold: (i) it establishes the topos as a foundational mathematical language for rigorously expressing the coupling between geometric representations and logical structures; (ii) it reveals deep connections between global network properties and local structural constraints; and (iii) it provides a rigorous theoretical basis for model interpretability, generalization analysis, and formal verification—thereby advancing machine learning toward provable correctness and logical reasoning. (128 words)

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Computer Vision: Other Foundations of Computer Vision

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

In this survey, we provide an overview of category theory-derived machine learning from four mainstream perspectives: gradient-based learning, probability-based learning, invariance and equivalence-based learning, and topos-based learning. For the first three topics, we primarily review research in the past five years, updating and expanding on the previous survey by Shiebler et al.. The fourth topic, which delves into higher category theory, particularly topos theory, is surveyed for the first time in this paper. In certain machine learning methods, the compositionality of functors plays a vital role, prompting the development of specific categorical frameworks. However, when considering how the global properties of a network reflect in local structures and how geometric properties are expressed with logic, the topos structure becomes particularly significant and profound.

Problem

• Category Theory
• Topology
• Machine Learning

Innovation

• Topological Data Analysis
• Machine Learning
• Geometric Logic