MINDS alumnus
Building stronger mathematical foundations for machine learning
Research focus: Category theory and the mathematical foundations of machine learning
Matthew Pugh is a MINDS alumnus whose doctoral research examined the mathematical foundations of machine learning, using category theory to describe how learning algorithms can be defined and constructed more rigorously. His work connects these questions with practical problems in sensing, anomaly detection and condition monitoring.
Career at a glance
| MINDS status | Alumnus |
| Background | Mechatronic engineering, University of Southampton |
| Doctoral project | Fault Prediction Using Digital Twin Techniques |
| Supervisors | Nick Harris, Jo Grundy and Mahesan Niranjan |
| Industry placements | Viper Innovations and Senseye |
| Research focus | Category theory and the mathematical foundations of machine learning |
Defining what it means to learn
The problem. Machine learning is used across science and industry, yet there is no single universally adopted mathematical definition covering everything described as machine learning. Deciding whether a method counts as machine learning often relies on convention and familiarity. That makes it harder to reason precisely about how an algorithm will behave and why.
The research. Matthew’s PhD began with anomaly prediction: spotting signs that an engineering system is drifting towards a fault. Working with real industrial data led him to a deeper question about how learning algorithms can be defined and built from first principles. His research used category theory to describe machine learning in precise mathematical terms.
Why the research matters
Industrial fault prediction needs decisions that can be explained, behaviour that can be relied on and clear accountability. Methods that are opaque find these requirements hard to meet. Stronger mathematical foundations could help in three ways.
Describing learning precisely
A formal account of learning may help researchers state what an algorithm assumes and what it can and cannot do.
Explaining model behaviour
Clearer foundations could support explanations of why a method behaves as it does, which matters where predictions inform safety decisions.
Connecting different methods
A shared mathematical framework may reveal links between algorithms that look different, and guide the design of new ones.
Research approach
Category theory is a branch of mathematics that studies structures and the relationships between them, across many different mathematical systems. Matthew used one of its tools, the Kan extension, to describe what learning from data involves.
In his framing, a dataset and a trained model can both be seen as ways of organising information under a set of rules. Training then becomes a structured transfer of data from one to the other. His published work shows that error minimisation problems, a core part of how many models learn, can be written in this form.
He also tested the approach on a practical problem. Using reasoning from Kan extensions, he designed an unsupervised anomaly detection method for a widely used occupancy sensor dataset, which performed strongly in the published evaluation.
Selected research outputs
Journal paper, 2025: Learning Is a Kan Extension. Matthew Pugh, Nick Harris, Corina Cirstea and Jo Grundy. Transactions on Machine Learning Research. A mathematical account of learning, showing how common training problems fit a single category-theory framework.
Journal paper, 2024: Using Kan extensions to motivate the design of a surprisingly effective unsupervised linear SVM on the occupancy dataset. Matthew Pugh, Jo Grundy, Corina Cirstea and Nick Harris. Mathematical and Computational Applications, 29(5), 74. Using the same mathematics to design an unsupervised method for spotting anomalies in sensor data.
From engineering problems to mathematical foundations
Matthew came to Southampton to study an integrated master’s in mechatronic engineering. During his third year, a conversation with his tutor, Nick Harris, made him aware of a MINDS PhD place in machine learning for anomaly prediction. He completed his degree at bachelor’s level and joined the programme.
His placements with Viper Innovations and Senseye gave him experience of real industrial data. He saw how far that data could fall from the tidy assumptions many methods rely on, and how much it matters to explain a model’s behaviour when its predictions inform safety decisions. Together with his reading of the literature, this shaped his move towards the mathematical foundations of machine learning.
Looking back
Connect with Matthew
Interested in mathematical approaches to machine learning, anomaly detection or sensor-based AI? Explore Matthew’s research or contact the MINDS team, who can direct a research or collaboration enquiry.