Sarkar, Rik

Bernabeu, Miguel O.

Andreeva, Rayna

2025-06-23T15:25:50Z

2025-06-23T15:25:50Z

2025-06-23

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.

https://hdl.handle.net/1842/43600

http://dx.doi.org/10.7488/era/6133

en

The University of Edinburgh

Rayna Andreeva, James Ward, Primoz Skraba, Jie Gao, and Rik Sarkar. Approximating metric magnitude of point sets. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 15374–15381, 2025

Rayna Andreeva, Katharina Limbeck, Bastian Rieck, and Rik Sarkar. Metric space magnitude and generalisation in neural networks. In Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), volume 221 of Proceedings of Machine Learning Research, pages 242–253. PMLR, 2023a

Rayna Andreeva, Benjamin Dupuis, Rik Sarkar, Tolga Birdal, and Umut Simsekli. Topological generalization bounds for discrete-time stochastic optimization algorithms. Advances in Neural Information Processing Systems, 37:4765–4818, 2024

Katharina Limbeck, Rayna Andreeva, Rik Sarkar, and Bastian Rieck. Metric space magnitude for evaluating the diversity of latent representations. Advances in Neural Information Processing Systems, 37:123911–123953, 2024

Rayna Andreeva, Anwesha Sarkar, and Rik Sarkar. Machine learning and topological data analysis identify unique features of human papillae in 3d scans. Scientific Reports, 13(1):21529, 2023b

Ameer Saadat-Yazdi, Rayna Andreeva, and Rik Sarkar. Topological detection of alzheimer’s disease using betti curves. In Interpretability of Machine Intelligence in Medical Image Computing, and Topological Data Analysis and Its Applications for Medical Data: 4th International Workshop, iMIMIC 2021, and 1st International Workshop, TDA4MedicalData 2021, Held in Conjunction with MICCAI 2021, Strasbourg, France, September 27, 2021, Proceedings 4, pages 119–128. Springer, 2021

Rayna Andreeva, Alessandro Fontanella, Ylenia Giarratano, and Miguel O Bernabeu. Dr detection using optical coherence tomography angiography (octa): a transfer learning approach with robustness analysis. In Ophthalmic Medical Image Analysis: 7th International Workshop, OMIA 2020, Held in Conjunction with MICCAI 2020, Lima, Peru, October 8, 2020, Proceedings 7, pages 11–20. Springer, 2020

Magnitude

Geometry

Topology

Machine Learning

Biomedical Data Analysis

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

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy