Machine Learning Becomes a Mathematical Collaborator | Quanta Magazine
Juhsz and Lackenby understood the strengths and weaknesses of machine learning. Given those, they hoped to use it to find novel connections between different types of invariants, which are properties used to distinguish knots from each other.
Two knots are considered different when its impossible to untangle them (without cutting them) so that they look like each other. Invariants are inherent properties of the knot that do not change during the untangling process (hence the name invariant). So if two knots have different values for an invariant, they can never be manipulated into one another.
There are many different types of knot invariants, characterized by how they describe the knot. Some are more geometric, others are algebraic, and some are combinatorial. However, mathematicians have been able to prove very little about the relationships between invariants from different fields. They typically dont know whether different invariants actually measure the same feature of a knot from multiple perspectives.
Juhsz and Lackenby saw an opportunity for machine learning to spot connections between different categories of invariants. From these connections they could gain a deeper insight into the nature of knot invariants.
Signature Verification
To pursue Juhsz and Lackenbys question, researchers at DeepMind developed a data set with over 2 million knots. For each knot, they computed different invariants. Then they used machine learning to search for patterns that tied invariants together. The computer perceived many, most of which were not especially interesting to the mathematicians.
We saw quite a few patterns that were either known or were known not to be true, said Lackenby. As mathematicians, we weeded out quite a lot of the stuff the machine learning was sending to us.
Unlike Juhsz and Lackenby, the machine learning system does not understand the underlying mathematical theory. The input data was computed from knot invariants, but the computer only sees lists of numbers.
As far as the machine learning system was concerned, these could have been sales records of various kinds of foods at McDonalds, said Davis.
Eventually the two mathematicians settled on trying to teach the computer to output an important algebraic invariant called the signature of a knot, based only on information about the knots geometric invariants.
After Juhsz and Lackenby identified the problem, researchers at DeepMind began to build the specific machine learning algorithm. They trained the computer to take 30 geometric invariants of a knot as an input and to output the knots signature. It worked well, and after a few weeks of work, DeepMind could accurately predict the signature of most knots.
Next, the researchers needed to find out how the model was making these predictions. To do this, the team at DeepMind turned to a technique known as saliency analysis, which can be used to tease out which of the many inputs are most responsible for producing the output. They slightly changed the value of each input, one at a time, and examined which change had the most dramatic impact on the output.
If an algorithm is designed to predict whether an image shows a cat, researchers performing saliency analysis will blur tiny sections of the picture and then check whether the computer still recognizes the cat. They might find, for instance, that the pixels in the corner of the image are less important than those that compose the cats ear.
When the researchers applied saliency analysis to the data, they observed that three of the 30 geometric invariants seemed especially important to how the model was making predictions. All three of these invariants measure features of the cusp, which is a hollow tube encasing the knot, like the rubber coating around a cable.
Based on this information, Juhsz and Lackenby constructed a formula which relates the signature of a knot to those three geometric invariants. The formula also uses another common invariant, the volume of a sphere with the knot carved out of it. When they tested the formula on specific knots, it seemed to work, but that wasnt enough to establish a new mathematical theorem. The mathematicians were looking for a precise statement that they could prove was always valid and that was harder.
It just wasnt quite working out, said Lackenby.
Juhsz and Lackenbys intuition, built up through years of studying similar problems, told them that the formula was still missing something. They realized they needed to introduce another geometric invariant, something called the injectivity radius, which roughly measures the length of certain curves related to the knot. It was a step that used the mathematicians trained intuition, but it was enabled by the particular insights they were able to glean from the many unedited connections identified by DeepMinds model.
The good thing is that [machine learning models] have completely different strengths and weaknesses than humans do, said Adam Zsolt Wagner of Tel Aviv University.
The modification was successful. By combining information about the injectivity radius with the three geometric invariants DeepMind had singled out, Juhsz and Lackenby created a failproof formula for computing the signature of a knot. The final result had the spirit of a real collaboration.
It was definitely an iterative process involving both the machine learning experts from DeepMind and us, said Lackenby.
Converting Graphs Into Polynomials
Building on the momentum of the knot theory project, in early 2020 DeepMind turned back to Williamson to see if he wanted to test a similar process in his field, representation theory. Representation theory is a branch of math that looks for ways of combining basic elements of mathematics like symmetries to make more sophisticated objects.
Within this field, Kazhdan-Lusztig polynomials are particularly important. They are based on ways of rearranging objects such as by swapping the order of two objects in a list called permutations. Each Kazhdan-Lusztig polynomial is built from a pair of permutations and encodes information about their relationship. Theyre also very mysterious, and it is often difficult to compute their coefficients.
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