Continuous Representations in Machine Learning - With applications to medical imaging and operator learning

Machine learning has led to significant advancements in many areas of our everyday lives, in addition to various fields in science. Machine and particularly deep learning methods have been used for applications such as protein structure prediction, drug discovery, climate modelling and medical imaging. Despite their successes, fundamental challenges remain regarding their theoretical understanding and interpretability, generalisation capabilities particularly in low data regimes, robustness to noise and training in restricted resource settings. Many real-world observations can naturally be described as continuous processes rather than discrete data points -- making their data inherently continuous. Consequently, in recent years, the deep learning research community has increasingly explored continuous representations, which refer to both continuous representations of data and neural networks. They enable resolution independent evaluation, as well as physics-informed and geometry-aware learning. In this thesis, the implications of continuous representations on open questions in machine learning research are investigated. We explore theoretical understanding, generalisation capabilities, robustness to noise, training in restricted resource settings and learning on arbitrary geometries. We show that continuous representations enable theoretical insights and improve interpretability by leveraging findings from well-established fields like ordinary differential equation theory. Through numerical experiments in the field of operator learning, we demonstrate that continuous representations improve generalisation, particularly in low data regimes, and provide theoretical and numerical insights into adaptive sampling strategies for continuous representations of neural networks. Furthermore, we demonstrate their robustness against noise, making them particularly well-suited for medical imaging, a field often challenged by significant noise levels. While neural networks are becoming increasingly more expressive and their number of parameters increases, their training and inference also becomes more and more challenging due to greater computational and memory costs. Thus, we propose a method to reduce memory and computational requirements during inference without the need to retrain the model. Additionally, owing to the resolution-invariant evaluation capability of continuous representations, we show that continuous representations enable efficient inference across varying resolutions. In the context of learning on irregular domains, we introduce a technique for operator learning on arbitrary geometries. As machine learning advances, continuous representations provide a powerful and flexible framework that addresses fundamental open questions in the field and promises to redefine current boundaries.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2025

Author dc:creator

• Runkel, Christina

Advisor dc:contributor.advisor

• Schoenlieb, Carola-Bibiane

• Continuous Representations
• Deep Learning
• Inverse Problems
• Machine Learning
• Medical Imaging
• Neural Networks
• Neural Operator
• Partial Differential Equations

Licence

• https://www.repository.cam.ac.uk/bitstreams/4ddca406-0bd2-42a0-81a1-ff3f38438fc3/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

Author Identifier

0009-0001-6980-9022

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/398241