Matthew Pugh MINDS CDT, 2019

Research Associate, University of Cambridge

Matthew Pugh is a MINDS alumnus now working as a Research Associate at the University of Cambridge. His route there was not straightforward and that turns out to be rather the point.

Matthew came to Southampton originally on an integrated masters in mechatronic engineering. In his third year, a chance conversation with his tutor Nick Harris opened up a different path: a PhD place in machine learning for anomaly prediction had become available, and the MINDS CDT had the resources to support it properly. He left his masters early with a bachelor's degree and joined the programme.

His PhD began in anomaly prediction but quickly moved somewhere less expected. The deeper he went into the literature, the more a particular gap became visible: there is no agreed, rigorous mathematical definition of what machine learning actually is. Identifying something as a machine learning algorithm is usually a "you know it when you see it" kind of exercise. That informality has consequences. Without precise foundations, it is harder to reason clearly about how algorithms behave, why they succeed or fail, and whether their outputs can be trusted.

That gap pulled Matthew toward category theory, an area of mathematics concerned with structure, relationships and abstraction at a very general level. The answer he arrived at reframes machine learning as a kind of data migration. A database stores information according to a set of rules, a formal theory. A machine learning model is, in this view, another kind of database one with more constraints, but fundamentally a mechanism for storing and organising data. Training a model is then a transfer between databases: migration from the original data into the structure the model defines. The category-theoretic formulation of this, in which machine learning algorithms are kan extensions of formal theories, makes the whole process mathematically precise.

His industry placements during the CDT, with Viper Innovations and Senseye pointed him in this direction as much as the theory did. Working with real industrial data, Matthew saw how far it fell from the clean assumptions that many ML approaches rely on. In settings where predictions inform decisions about safety and people's livelihoods, the ability to explain and justify a model's behaviour is not optional. The experience convinced him that robust foundations were not an academic indulgence but a practical necessity.

At Cambridge, Matthew is working on the AI for Mathematics grant, investigating how AI can be used to automatically write proofs for mathematical conjectures. The connection to his PhD is direct: the category theory and machine learning background he built at Southampton is precisely the combination the research group needed. The tools being developed have implications beyond pure mathematics, with potential applications in cybersecurity, hardware design and the ability of AI agents to plan and reason reliably.

Alongside his research, Matthew has co-founded UpskillAI, a consulting and education company that works with organisations to bridge the gap between the real-world problems they face and the machine learning expertise that can address them. It reflects a conviction he has carried since his placement years: that research only matters if its knowledge can reach the people who need it.

How high you can build depends on the depth of your foundations.