A Neural Network for Semigroups
This work addresses a niche algebraic problem for researchers in semigroup theory, with incremental contributions to neural network applications in mathematics.
The paper tackled the problem of completing partial multiplication tables of finite semigroups using a denoising autoencoder-based neural network with a novel loss function, achieving about 80% reconstruction accuracy from half the table with only 10% of the data.
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 package for conducting experiments similar to those carried out in this work. Our experiments showed that with only about 10% of the available data, it is possible to build a model capable of reconstructing a full Cayley from only half of it in about 80% of cases.