Learning to be Simple

Conjectures have historically played an important role in the development of pure mathematics. We propose a systematic approach to finding abstract patterns in mathematical data, in order to generate conjectures about mathematical inequalities, using machine intelligence. We focus on strict inequalities of type f < g and associate them with a vector space. By geometerising this space, which we refer to as a conjecture space, we prove that this space is isomorphic to a Banach ...

When trying to fit a deep neural network (DNN) to a $G$-invariant target function with $G$ a group, it only makes sense to constrain the DNN to be $G$-invariant as well. However, there can be many different ways to do this, thus raising the problem of ``$G$-invariant neural architecture design'': What is the optimal $G$-invariant architecture for a given problem? Before we can consider the optimization problem itself, we must understand the search space, the architectures in ...

We introduce a method to design a computationally efficient $G$-invariant neural network that approximates functions invariant to the action of a given permutation subgroup $G \leq S_n$ of the symmetric group on input data. The key element of the proposed network architecture is a new $G$-invariant transformation module, which produces a $G$-invariant latent representation of the input data. This latent representation is then processed with a multi-layer perceptron in the net...

The Kronecker coefficients are the decomposition multiplicities of the tensor product of two irreducible representations of the symmetric group. Unlike the Littlewood--Richardson coefficients, which are the analogues for the general linear group, there is no known combinatorial description of the Kronecker coefficients, and it is an NP-hard problem to decide whether a given Kronecker coefficient is zero or not. In this paper, we show that standard machine-learning algorithms ...

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for groups and practically by providing succinct means of representing group elements. We give a survey of results obtained in the study of these symmetric generating sets. In keeping with earlier surveys on this matter, we emphasize the sporadic si...

Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this paper, we seek to extend these techniques to finitely presented non-free groups, with a particular emphasis on polycyclic and metabelian groups that are of interest to non-commutative cryptography. As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., deter...

We survey some recent applications of machine learning to problems in geometry and theoretical physics. Pure mathematical data has been compiled over the last few decades by the community and experiments in supervised, semi-supervised and unsupervised machine learning have found surprising success. We thus advocate the programme of machine learning mathematical structures, and formulating conjectures via pattern recognition, in other words using artificial intelligence to hel...

Recent work has applied supervised deep learning to derive continuous symmetry transformations that preserve the data labels and to obtain the corresponding algebras of symmetry generators. This letter introduces two improved algorithms that significantly speed up the discovery of these symmetry transformations. The new methods are demonstrated by deriving the complete set of generators for the unitary groups U(n) and the exceptional Lie groups $G_2$, $F_4$, and $E_6$. A thir...

The purpose of this article is to review the achievements made in the last few years towards the understanding of the reasons behind the success and subtleties of neural network-based machine learning. In the tradition of good old applied mathematics, we will not only give attention to rigorous mathematical results, but also the insight we have gained from careful numerical experiments as well as the analysis of simplified models. Along the way, we also list the open problems...

We propose a computationally efficient $G$-invariant neural network that approximates functions invariant to the action of a given permutation subgroup $G \leq S_n$ of the symmetric group on input data. The key element of the proposed network architecture is a new $G$-invariant transformation module, which produces a $G$-invariant latent representation of the input data. Theoretical considerations are supported by numerical experiments, which demonstrate the effectiveness and...