Metric magnitude and topological methods for machine learning and biomedical data analysis

Abstract

We live in a world which generates vast amounts of data with highly complex structure. Methods based on geometry and topology are suited to analyse the shape of high-dimensional data and thus can provide unique insights. While geometry is concerned with studying distances, topology focuses on connectivity relations. The main advantage of these methods is that they can generate compact summaries of the data to highlight and unravel distinct patterns and relationships. Magnitude is a recently introduced geometric invariant, capable of capturing important properties of the intrinsic geometry of a space. It has potential for applications in machine learning as it can measure a number of geometric quantities such as curvature, volume and diameter. In this thesis, we provide the first applications of magnitude to theoretical deep learning, representation learning and biomedical data analysis. In addition, we compare the geometric insights from magnitude with the topological insights from persistent homology. This thesis contains three parts, the first addresses one of the main difficulties in the application of magnitude, which is the computational cost. To compute magnitude, one needs to invert a matrix, which is an expensive procedure, particularly for large datasets. We provide new faster algorithms for speeding up this computation and approximate magnitude well. These new algorithms enable the applicability of magnitude to data analysis, providing a solid foundation for its wider adoption. The second part examines the intrinsic geometric aspect of machine learning. Here we show the unique uses of magnitude to generalization and the space of latent representations. In the third part, we demonstrate novel biomedical applications of magnitude to the surface of the human tongue and brain artery trees.