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Meet the JRF: Dr Matthew Colbrook explains how maths can help artificial intelligence

Trinity Junior Research Fellow in Mathematics, Dr Matthew Colbrook, discusses his interest in artificial intelligence and how mathematics can increase our understanding of AI

Tell us about a key research interest

A vast number of problems in science and engineering involve an infinite amount of information. However, we can only perform computations using a finite computer that executes a finite number of operations on a finite amount of data. How can we reliably approximate an infinite problem by a finite one? An ‘infinite’ approach to numerical analysis is often crucial to answering this question. My research marries such an approach with the mathematics of information to solve challenging computational problems. Furthermore, I classify the difficulty of problems and prove that algorithms are optimal for a given problem.

For example, spectral computations – a way of splitting up complex systems into simpler ones – are ubiquitous in the sciences. However, their many applications and theoretical studies depend on notoriously difficult computations. Part of my work figures out the boundaries of what computers can achieve in computational spectral theory and, as a consequence, physics. For example, physicists can now use these new techniques to compute energy levels of quantum systems whilst controlling the error of the computation, and have even used them to verify the discovery of a new type of quantum state in quasicrystals.

How does this work relate to artificial intelligence?

Neural networks, the state-of-the-art tool in AI, roughly mimic the links between neurons in the brain. I have applied the above framework to uncover intrinsic barriers in training (from data) stable and accurate neural networks.

It is hard to overstate the impact of neural networks and AI over the last decade. Because of AI, we now have smart speakers in our homes, driving assistance, and automated medicine diagnosis etc. Yet, there is also overwhelming empirical evidence that modern AI is often unstable (non-robust), can hallucinate (see, for example, the hallucinations in Facebook and NYU’s fastMRI challenge for medical image reconstruction), and can thus produce nonsensical output whilst being highly confident in its results.

Stability means that a small change to the input will only lead to a small change in the output. For example, if a patient moves slightly during an MRI scan, you want the image reconstruction to remain stable. Instabilities can be very dangerous in safety-critical applications and can also be very difficult for AI to detect. The European Commission for Legal AI has raised these issues as a serious concern.

My research identifies the following paradox: there are problems where stable and accurate neural networks exist, yet no algorithm can produce such a network.

The impossibility of computing the good existing neural network is also true regardless of the amount of training data. No matter how much data an algorithm can access, it will not produce the desired network. Only in specific cases can algorithms compute stable and accurate neural networks. We propose a classification theory describing when neural networks can be trained to provide a trustworthy AI system under certain specific conditions.

Why do you find this compelling?

The above paradox has its roots in two mathematical giants: Alan Turing and Kurt Gödel. At the beginning of the twentieth century, mathematicians attempted to justify mathematics as the ultimate consistent language of the universe. However, Turing and Gödel showed a paradox at the heart of mathematics: it is impossible to prove whether certain mathematical statements are true or false, and some computational problems cannot be tackled with algorithms.

Moreover, whenever a mathematical system is rich enough to describe the arithmetic we learn at school, it cannot prove its own consistency. I find it fascinating that similar paradoxes are now appearing in AI.

It is essential to understand the limitations of different approaches. We are at the stage where the practical successes of AI are far ahead of theory and understanding. A research programme on understanding the foundations of AI computing is needed to bridge this gap. Just as the paradoxes on the limitations of mathematics and computers identified by Gödel and Turing led to rich foundation theories – describing both the limitations and the possibilities of mathematics and computations – perhaps a similar foundations theory may blossom in AI.

Should we be wary about AI systems?

AI is not inherently flawed, but it is only reliable in specific areas, using specific methods. The issue is with areas where you need a guarantee on the correctness of the results. It is fine in some situations for AI to make mistakes or potentially be unstable, but it needs to be honest about it. And that is not the case for many current systems.

When twentieth-century mathematicians identified different paradoxes, they did not stop studying mathematics. Instead, they just had to find new paths because they understood the limitations. For AI, it may be a case of changing paths or developing new routes to build systems that can solve problems in a trustworthy and transparent way while understanding their limitations.

Is the excitement about AI justified?

Yes and no. Yes, because AI has already made great strides in many areas. There are many problems where it achieves things that we could only have dreamed of a decade ago, and this will continue to be the case in the coming decade. No, because it can sometimes be oversold, like any new scientific area or new technology.

The strong optimism surrounding AI is comparable to the optimism surrounding mathematics in the early twentieth century led by David Hilbert. He sought a research programme on the foundations of mathematics, believing that mathematics could prove or disprove any statement and, moreover, that there were no restrictions on which problems could be solved by algorithms. The seminal contributions of Gödel and Turing described above turned Hilbert’s optimism upside down, establishing paradoxes leading to impossibility results on what mathematics and digital computers can achieve. A programme on the boundaries and limitations of AI, similar to Hilbert’s programme, is needed.

What’s the alternative?

I am not sure it is a question of finding an alternative to AI. Instead, we must figure out which problems cannot be safely tackled with AI or need an alternative way of using AI (in some applications, stability is utterly essential). There is absolutely no doubt that AI has many things to offer.

The results of my research are not all negative. There are two sides to figuring out the foundations of AI. One shows methodological barriers, and the second creates algorithms that achieve the boundaries of what is possible, ie optimal algorithms. This tells us which directions we can take with the exciting new AI technology. Figuring out what can and cannot be done will be healthy for AI in the long run. Moreover, often such a programme leads to new techniques and methodology.

What do you hope to achieve as a JRF and how are you finding life Trinity?

I have two main aims. The first is to develop my research area so that after my time as a JRF I can build a successful team of researchers. There is much to be done in building a numerical analysis and foundations programme in AI, infinite-dimensional spectral theory, and other areas such as PDEs and optimisation. Secondly I am enjoying being an active member of Trinity, including discussions with other Fellows and initiatives such as the Postdoc-PhD Mentoring Programme. Research is not something done in isolation, and it is fantastic to be part of Trinity’s supportive community.

How important is it to promote the importance of mathematics in society and how best to go about that?

Mathematics underpins so much of what society takes for granted. Mathematics has always stood behind key technological and scientific advances: from the time of the ancient Greeks, through to breakthroughs in physics, chemistry, and biology in the twentieth century; and how we tackle issues such as the COVID pandemic today.

The importance of mathematics will only increase as our world becomes more data-driven. We should also promote the role of mathematics in guiding us to figure out what is and what is not possible, and how to tackle those problems that can be solved.

My view is that there are two related ways to promote mathematics in the wider world. The first, which applies to any subject, is education from an early age. There are substantial educational disparities between people of different socio-economic backgrounds, now magnified by the pandemic, which must be addressed.

Mathematics could and should be made more fun and engaging at school, with students encouraged how to think creatively to solve a problem.

The second way to promote mathematics is to demonstrate what it enables us to do in our daily lives, as well as getting rid of the fear factor. A basic level of mathematics and understanding of its importance should be accessible to all. As mathematicians, we sometimes like to exist in our own bubble. The onus is on us to break down this barrier and engage with the broader public.


Read Dr Colbrook’s co-authored paper: ‘The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale’s 18th problem.’

Dr Coolbrook received the SIAM (Society for Industrial and Applied Mathematics) Richard C Di Prima Prize 2022. Read more: Honours and awards for Trinity Fellows – Trinity College Cambridge

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