A Neural Network for Semigroups

TL;DR

This paper introduces a neural network approach using a denoising autoencoder and a novel loss function to reconstruct finite semigroup multiplication tables from partial data, demonstrating high accuracy with limited information.

Contribution

It presents a new neural network architecture and loss function tailored for algebraic data completion, specifically for finite semigroups, with supporting software and experimental validation.

Findings

Reconstructed full Cayley tables with 80% accuracy from 50% data

Achieved successful reconstruction with only 10% of the data available

Provided a software package for similar algebraic data experiments

Abstract

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.