Mathematical Foundations of Intelligence: An "Erlangen Programme" for AI

In 1872, Felix Klein published his now famous Erlangen Programme, in which he treated geometry as the study of invariants, formalised using group theory. This radically new approach allowed tying together different types of non-Euclidean geometries that had emerged in the nineteenth century and has had a profound methodological and cultural impact on geometry in particular and mathematics in general. New fields of mathematics such as exterior calculus, algebraic topology, the theory of fibre bundles and sheaves, and category theory emerged as a continuation of Klein's blueprint. The Erlangen Programme was also fundamental for the development of physics in the first half of the twentieth century, with Noether's theorem and the notion of gauge invariance successfully providing a unification framework for electromagnetic, weak, and strong interactions, culminating in the Standard Model in the 1970s. Now is the time for an "Erlangen Programme" for AI, based on rigorous mathematical principles that would bring better understanding of existing AI methods as well as a new generation of methods that have guaranteed expressive and generalisation power, better interpretability, scalability, and data- and computational-efficiency. Just as the ideas of Klein's Erlangen Programme spilled into other disciplines and produced new theories in mathematics, physics, and beyond, we will draw inspiration from these analogies in our AI research programme. By resorting to powerful tools from the mathematical and algorithmic fields sometimes considered "exotic" in applied domains, new theoretical insights and computational models can be derived. Our "Erlangen Programme of AI" will study four fundamental questions that underlie modern AI/ML systems, striving to provide rigorous mathematical answers. How can hidden structures in data be discovered and expressed in the language of geometry and topology in order to be exploited by ML models? Can we use geometric and topological tools to characterise ML models in order to understand when and how they work and fail? How can we guarantee learning to benefit from these structures, and use these insights to develop better, more efficient, and safer new models? Finally, how can we use such models in future AI systems that make decisions potentially affecting billions of people? With a centre at Oxford, and broad geographic coverage of the UK, the Hub will bring together leading experts in mathematical, algorithmic, and computational fields underpinning AI/ML systems as well as their applications in scientific and industrial settings. Some of the Hub participants have a track record of previous successful work together, while other collaborations are new. The research programme in the proposed Hub is intended to break barriers between different fields and bring a diverse and geographically-distributed cohort of leading UK experts rarely seen together with the purpose of strong cross-fertilisation. In the fields of AI/ML, our work will contribute to the exploitation of tools from currently underexplored mathematical fields. Conversely, our programme will help attract the attention of theoreticians to new problems and applications.