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AI discovered two major mathematical conjectures on the cover of Nature, and the hero behind it was DeepMind!
This time, machine learning is the first to discover mathematical connections that humans have overlooked.
The practice of mathematics is simply to discover certain patterns and use these patterns to propose and prove conjectures, thus forming theorems.
Recently, DeepMind used machine learning to help mathematicians discover new conjectures and theorems in the fields of Knot theory and Representation theory.
Mathematicians have been using computers to help discover patterns and conjectures since the 1960s, most notably the Birch and Swinnerton-Dyer conjecture, one of the millennium prize conundrums.
To discover potential patterns and associations between mathematical objects, the DeepMind research team proposed a new machine learning model that uses attribution techniques to aid understanding and use these observations to further guide the process of intuitive thinking and conjecture.
The paper "Advancing mathematics by guiding human intuition with AI" was published in Nature on December 1.
Low-dimensional topology is an active and influential field of mathematics.
Kinks, which are simple closed curves in three-dimensional space, are also one of the basic objects in low-dimensional topologies.
In mathematical language, a knot is the embedding of a circle in a 3-dimensional Euclidean space. If two mathematical knots can be transformed into another knot by the deformation of R, then the two knots are equivalent.
The researchers found that the "Saliency map" technique commonly used in the field of computer vision (CV) could be used in this problem. In the CV task, the technique can determine which parts of the image carry the most relevant information.
With this technique, the computer gives properties that may be related in multiple knots and generates a formula that seems to work in all cases.
One of the methods for the study of kinks is achieved through invariants, which are algebraic, geometric, or numerical quantities that are identical to any two equivalent knots.
These invariants are derived in many different ways, and the paper focuses on two of the main categories: hyperbolic invariants and algebraic invariants. These two classes of invariants are derived from quite different mathematical disciplines, so it makes sense to make connections between them.
An example of a notable conjecture link is the volume conjecture, which proposes that a knot's hyperbolic volume (geometric invariant) should be encoded in the asymptotic behavior of its coloured Jones polynomials (algebraic invariants).
Example of invariants
The researchers hypothesized that there was an undiscovered relationship between a knotted hyperbolic invariant and an algebraic invariant.
Thus, DeepMind discovered a direct link between the geometric invariants of a node and a specific number of generations, the signature σ(K), which was previously completely unknown and could not be hinted at by existing theories.
The three most relevant features of invariants are identified using attribution techniques and partially visualized
In addition, by using attribution techniques for machine learning, DeepMind introduced a new quantity " natural slope " defined as slip(K) = Re(λ/μ), where Re represents the real part.
Conjecture: There are constants c1 and c2, such that for each hyperbolic knot K.
This conjecture is supported by the analysis of several large datasets sampled from different distributions.
Theorem: There is a constant c such that for any hyperbolic knot K.
It turns out that this formula applies even for a very large class of knots.
In the entire dataset generated, c≥ 0.23392 can be used as the lower bound, and it is reasonable to guess that c is at most 0.3, which gives a close relationship within the calculated region.
It's amazing that in a field that has been extensively studied, a simple and profound connection like this has been overlooked.
Mark Brittenham, a Knot theorist at the University of Nebraska-Lincoln, said: "This article proves the correlation of these invariants in a very direct way, which shows that there are still some fundamental things in this field that we do not fully understand."
Representation theory "represents" the elements of abstract algebraic structures as linear transformations on vector space and studies the modulus of these algebraic structures to study the properties of structures. At the same time, representation theory is also a theory of linear symmetry.
The problem of "symmetry" in the transformation of a finite set of objects is of great significance in several branches of mathematics and has long been a hot topic of study by mathematicians, using tools such as graphs and polynomials.
Professor Geordie Williamson of the University of Sydney said researchers had been hoping for decades that polynomials might be calculated from networks, but that goal seemed elusive.
Graphs become too large and complex, "soon beyond human comprehension."
Now, with the help of computers, he and his team noticed that it was possible to break down the diagram into smaller, more manageable parts, each with a high-dimensional cube structure.
Williamson said he was deeply shocked by the power of AI. Once a machine learning algorithm locks in a pattern, it can very accurately guess which graphs and polynomials come from the same symmetry, and guess so accurately!
For nearly 40 years, progress has been made on the combinatorial invariance conjecture, which states that the KL polynomials of two elements in a symmetry group SN can be computed from their unlabeled Bruhat interval, i.e. a directed graph.
Irreducible representations are influenced by Kazhdan-Lusztig (KL) polynomials, which are deeply related to combinatorics, algebraic geometry, and singularity theory.
Among them, the Bruhat interval is a chart that represents all the different ways to reverse the order of the set of objects by swapping only two objects at a time. The KL polynomial tells mathematicians something about the different ways in which this graph exists in high-dimensional space. Interesting structures only start to appear when bruhat intervals have 100 or 1000 vertices.
One obstacle to progress in understanding the relationship between these objects is that the Bruhat intervals of non-third-order KL polynomials (those not equal to 1) are very large graphs that are difficult to form intuitive understanding.
Using this conjecture as an initial hypothesis, DeepMind discovered a supervised learning model capable of predicting the Bruhat interval with fairly high precision of KL polynomials.
By varying the way Bruhat intervals are fed into the neural network, DeepMind found that some choices of graphs and features were particularly beneficial to prediction accuracy. Especially with the support of more accurate estimation functions, it is enough to compute the KL polynomial with some subgraphs.
The most relevant subgraphs can be identified by computational attribution techniques and the marginal distribution of these plots with the original plots is analyzed to uncover further structural evidence.
In Figure a below, DeepMind summarizes the relative frequencies of edges in the subgraph by "reflection".
The results show that the extreme reflection (SN in the form [0,i] or [i,N-1]) appears more often in the most relevant subgraphs, while the simple reflection (in the form [i,i+1]) is discarded, which has been confirmed by repeated model retraining.
From the definition of KL polynomials, the relationship between the difference between simple reflections and extreme reflections and the correlation of subgraphs is intuitive. With this in mind, DeepMind found that a Bruhat interval can be naturally decomposed into two parts: a hypercube induced by a set of extreme edges and a diagram isomorphic to the interval in SN-1.
Theorem: Each Bruhat interval has a typical hypercube decomposition that is reflected along its extremum, from which the KL polynomial can be directly calculated.
Notably, further testing showed that all hypercube decomposition can determine the KL polynomial. This is computationally verified in all 3×106 intervals in the symmetry group below S7 and 1.3 ×105 intervals in the non-isomorphic intervals of S8 and S9.
Conjecture: The KL polynomial of the unlabeled Bruhat interval can be used to calculate any hypercube decomposition using the preceding formula.
If this conjecture can be proved, then the conjecture of combinatorial invariance of symmetric groups can be solved.
More than a century ago, Srinivasa Ramanujan shocked the mathematical community with her ability to see extraordinary patterns in numbers that no one else could see. The self-taught mathematician from India describes his insight as "deep intuition and spirit".
It is well known that the intuition of mathematicians plays an extremely important role in mathematical discovery – only by combining rigorous formalism and good intuitive thinking can complex mathematical problems be solved.
Centuries ago, a relationship between convex polyhedra properties was discovered: regardless of shape, the number of vertices (v) minus the number of edges (e) plus the number above (f) equals 2, i.e+ f = 2.
The formula that describes this relationship is called Euler's formula, named after the Swiss mathematician Leonhard Euler, and is known as the God formula.
Mathematicians generally develop mathematical theories by studying examples in a cyclical manner. In addition, creativity and numeracy are required.
In this simple problem case, they can study several examples of different shapes with paper and pen to derive this formula.
But when you encounter more complex mathematical problems, you need a wider range of calculations, and you have to need machine learning to help.
Beginning in the 1960s, mathematicians began to use computers to help discover laws and come up with conjectures, but AI systems have not been widely used in theoretical mathematical research.
This universal machine learning framework approach, described by DeepMind in his paper, allows mathematicians to use ML tools to guide their intuition about complex mathematical objects, to verify assumptions about the existence of relationships, and to understand those relationships.
By training a machine learning model to estimate a function on the PZ distribution of a particular data, the process helps guide mathematicians' intuitive thinking about the hypothesis function f.
Insights into the accuracy of the learning function fˆ and the attribution techniques applied to it can help understand the problem and construct a closed form of f'.
At the same time, the process is iterative and interactive, not a series of steps.
The results of the study prove that, guided by the intuitive thinking of mathematicians, machine learning provides a powerful framework for discovering interesting and provable conjectures in areas where large amounts of data are available, or where objects are too large to apply classical methods to study.
As the researchers conclude, "Intuition plays an important role in many of the paranormal manifestations that humans seek."
For example, it's crucial for top Go players, and AlphaGo has succeeded in part because it uses machine learning to learn the ability of humans to intuitively judge elements of the game.
Mathematics is indeed a very different and more cooperative work from Go, so AI does have the space and potential to be effective in assisting mathematicians in completing related aspects.
Marc Lackenby, one of the mathematicians involved in the study, from the University of Oxford in the United Kingdom, said: "I was struck by how useful machine learning tools are. I didn't expect some of my preconceived notions to be subverted."
Mathematician Jeffrey Weeks says he has pioneered some of these techniques since the 1980s. But the use of AI to let computers find patterns has given this research a qualitative boost.
Resources:
https://www.nature.com/articles/s41586-021-04086-x https://www.nature.com/articles/d41586-021-03593-1 https://deepmind.com/blog/article/exploring-the-beauty-of-pure-mathematics-in-novel-ways