Grokking Group Multiplication with Cosets

We introduce a machine-learning approach (denoted Symmetry Seeker Neural Network) capable of automatically discovering discrete symmetry groups in physical systems. This method identifies the finite set of parameter transformations that preserve the system's physical properties. Remarkably, the method accomplishes this without prior knowledge of the system's symmetry or the mathematical relationships between parameters and properties. Demonstrating its versatility, we showcas...

This paper presents a neurosymbolic approach to classifying Galois groups of polynomials, integrating classical Galois theory with machine learning to address challenges in algebraic computation. By combining neural networks with symbolic reasoning we develop a model that outperforms purely numerical methods in accuracy and interpretability. Focusing on sextic polynomials with height $\leq 6$, we analyze a database of 53,972 irreducible examples, uncovering novel distribution...

Understanding the internal representations learned by neural networks is a cornerstone challenge in the science of machine learning. While there have been significant recent strides in some cases towards understanding how neural networks implement specific target functions, this paper explores a complementary question -- why do networks arrive at particular computational strategies? Our inquiry focuses on the algebraic learning tasks of modular addition, sparse parities, and ...

We consider the problem of learning a function respecting a symmetry from among a class of symmetries. We develop a unified framework that enables symmetry discovery across a broad range of subgroups including locally symmetric, dihedral and cyclic subgroups. At the core of the framework is a novel architecture composed of linear, matrix-valued and non-linear functions that expresses functions invariant to these subgroups in a principled manner. The structure of the architect...

We introduce the HyperCube network, a novel approach for autonomously discovering symmetry group structures within data. The key innovation is a unique factorization architecture coupled with a novel regularizer that instills a powerful inductive bias towards learning orthogonal representations. This leverages a fundamental theorem of representation theory that all compact/finite groups can be represented by orthogonal matrices. HyperCube efficiently learns general group oper...

The lack of interpretability and trust is a much-criticised feature of deep neural networks. In fully connected nets, the signalling between inner layers is scrambled because backpropagation training does not require perceptrons to be arranged in any particular order. The result is a black box; this problem is particularly severe in scientific computing and digital signal processing (DSP), where neutral nets perform abstract mathematical transformations that do not reduce to ...

Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We prese...

Grokking has been actively explored to reveal the mystery of delayed generalization. Identifying interpretable algorithms inside the grokked models is a suggestive hint to understanding its mechanism. In this work, beyond the simplest and well-studied modular addition, we observe the internal circuits learned through grokking in complex modular arithmetic via interpretable reverse engineering, which highlights the significant difference in their dynamics: subtraction poses a ...

Machine learning explorations can make significant inroads into solving difficult problems in pure mathematics. One advantage of this approach is that mathematical datasets do not suffer from noise, but a challenge is the amount of data required to train these models and that this data can be computationally expensive to generate. Key challenges further comprise difficulty in a posteriori interpretation of statistical models and the implementation of deep and abstract mathema...

Why does Deep Learning work? What representations does it capture? How do higher-order representations emerge? We study these questions from the perspective of group theory, thereby opening a new approach towards a theory of Deep learning. One factor behind the recent resurgence of the subject is a key algorithmic step called pre-training: first search for a good generative model for the input samples, and repeat the process one layer at a time. We show deeper implications ...