May 2, 2019
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March 13, 2022
Along with the proliferation of digital data collected using sensor technologies and a boost of computing power, Deep Learning (DL) based approaches have drawn enormous attention in the past decade due to their impressive performance in extracting complex relations from raw data and representing valuable information. Meanwhile, though, rooted in its notorious black-box nature, the appreciation of DL has been highly debated due to the lack of interpretability. On the one hand,...
March 12, 2021
Tasks like image reconstruction in computer vision, matrix completion in recommender systems and link prediction in graph theory, are well studied in machine learning literature. In this work, we apply a denoising autoencoder-based neural network architecture to the task of completing partial multiplication (Cayley) tables of finite semigroups. We suggest a novel loss function for that task based on the algebraic nature of the semigroup data. We also provide a software packag...
April 15, 2024
What has an Artificial Neural Network (ANN) learned after being successfully trained to solve a task - the set of training items or the relations between them? This question is difficult to answer for modern applied ANNs because of their enormous size and complexity. Therefore, here we consider a low-dimensional network and a simple task, i.e., the network has to reproduce a set of training items identically. We construct the family of solutions analytically and use standard ...
November 13, 2023
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 ...
April 21, 2022
We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised learning. Additionally, unsupervised methods can provide insight into the structure of such geometrical data. At the heart of this programme is the question of how geometry can be machine learned, and indeed how AI helps one to do mathematics...
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...
November 25, 2021
Is intelligence realized by connectionist or classicist? While connectionist approaches have achieved superhuman performance, there has been growing evidence that such task-specific superiority is particularly fragile in systematic generalization. This observation lies in the central debate between connectionist and classicist, wherein the latter continually advocates an algebraic treatment in cognitive architectures. In this work, we follow the classicist's call and propose ...
December 11, 2023
This paper explores the application of automated planning to automated theorem proving, which is a branch of automated reasoning concerned with the development of algorithms and computer programs to construct mathematical proofs. In particular, we investigate the use of planning to construct elementary proofs in abstract algebra, which provides a rigorous and axiomatic framework for studying algebraic structures such as groups, rings, fields, and modules. We implement basic i...
March 27, 2016
There exists a theory of a single general-purpose learning algorithm which could explain the principles of its operation. This theory assumes that the brain has some initial rough architecture, a small library of simple innate circuits which are prewired at birth and proposes that all significant mental algorithms can be learned. Given current understanding and observations, this paper reviews and lists the ingredients of such an algorithm from both architectural and function...
April 1, 1992
In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion...