Machine Learning with Geometric Algebra: Multivectors for Modelling, Understanding and Computing

Abstract

Geometric Algebra (GA) has been successfully applied in several fields, including physics, graphics, and robotics, but its potential in Machine Learning (ML) and Deep Learning (DL) remains largely unexplored. This thesis addresses that gap by investigating the application of GA to a variety of ML tasks that are inherently geometric in nature. Our premise is simple yet powerful: if, as it is often said, ML is a clever rebranding of linear algebra, then geometric problems in ML deserve to be tackled with GA, an extension of linear algebra designed to represent geometric objects and perform transformations on them naturally and compactly. The GA framework provides a unified language to represent, understand, and manipulate geometric entities using multivectors, rotors, and sandwich products. The result is a set of tools that are not only mathematically elegant but also highly effective in practice, spanning a broad range of domains: Chapter 2 tackles regression tasks on rotation groups and molecular geometry optimization; Chapter 3 explores protein modelling and structure prediction; Chapter 4 addresses 3D camera pose estimation and 3D line alignment; Chapter 5 focuses on the solution of partial differential equations for computational fluid dynamics and electromagnetism. We demonstrate that GA can be employed as a versatile, practical and principled framework for building geometry-aware ML systems independently of the employed architecture. By embedding geometric priors directly into the model architecture via GA, we unlock several advantages, including lower regression errors, robustness to noise and transformations and interpretability of intermediate computations. More broadly, this thesis is an invitation to rethink how neural networks should represent, model, and transform geometric data, grounding these operations in an algebra that reflects the structure of the problem itself. As ML continues to engage with increasingly complex and structured data, the need for such expressive representations will only grow, and GA offers a compelling framework to meet that need.