December 8, 2023
In this work we employ machine learning to understand structured mathematical data involving finite groups and derive a theorem about necessary properties of generators of finite simple groups. We create a database of all 2-generated subgroups of the symmetric group on n-objects and conduct a classification of finite simple groups among them using shallow feed-forward neural networks. We show that this neural network classifier can decipher the property of simplicity with varying accuracies depending on the features. Our neural network model leads to a natural conjecture concerning the generators of a finite simple group. We subsequently prove this conjecture. This new toy theorem comments on the necessary properties of generators of finite simple groups. We show this explicitly for a class of sporadic groups for which the result holds. Our work further makes the case for a machine motivated study of algebraic structures in pure mathematics and highlights the possibility of generating new conjectures and theorems in mathematics with the aid of machine learning.
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We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we find that questions such as identification of simple groups by "looking" at the Cayley table or correctly matching addition and multiplication tables for finite rings can, at least for structures of small size, be performed by the AI, even...
February 6, 2023
Universality is a key hypothesis in mechanistic interpretability -- that different models learn similar features and circuits when trained on similar tasks. In this work, we study the universality hypothesis by examining how small neural networks learn to implement group composition. We present a novel algorithm by which neural networks may implement composition for any finite group via mathematical representation theory. We then show that networks consistently learn this alg...
January 15, 2021
We review, for a general audience, a variety of recent experiments on extracting structure from machine-learning mathematical data that have been compiled over the years. Focusing on supervised machine-learning on labeled data from different fields ranging from geometry to representation theory, from combinatorics to number theory, we present a comparative study of the accuracies on different problems. The paradigm should be useful for conjecture formulation, finding more eff...
December 11, 2023
We use the group Fourier transform over the symmetric group $S_n$ to reverse engineer a 1-layer feedforward network that has "grokked" the multiplication of $S_5$ and $S_6$. Each model discovers the true subgroup structure of the full group and converges on circuits that decompose the group multiplication into the multiplication of the group's conjugate subgroups. We demonstrate the value of using the symmetries of the data and models to understand their mechanisms and hold u...
December 20, 2014
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 ...
September 11, 2023
We consider the problem of discovering subgroup $H$ of permutation group $S_{n}$. Unlike the traditional $H$-invariant networks wherein $H$ is assumed to be known, we present a method to discover the underlying subgroup, given that it satisfies certain conditions. Our results show that one could discover any subgroup of type $S_{k} (k \leq n)$ by learning an $S_{n}$-invariant function and a linear transformation. We also prove similar results for cyclic and dihedral subgroups...
In this work, we formally prove that, under certain conditions, if a neural network is invariant to a finite group then its weights recover the Fourier transform on that group. This provides a mathematical explanation for the emergence of Fourier features -- a ubiquitous phenomenon in both biological and artificial learning systems. The results hold even for non-commutative groups, in which case the Fourier transform encodes all the irreducible unitary group representations. ...
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 {\em pretraining}: first search for a good generative model for the input samples, and repeat the process one layer at a time. We show deeper implicat...
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...
There are two big unsolved mathematical questions in artificial intelligence (AI): (1) Why is deep learning so successful in classification problems and (2) why are neural nets based on deep learning at the same time universally unstable, where the instabilities make the networks vulnerable to adversarial attacks. We present a solution to these questions that can be summed up in two words; false structures. Indeed, deep learning does not learn the original structures that hum...