Predicting the cardinality and maximum degree of a reduced Gr\"obner basis

TL;DR

This paper develops neural network models to predict the complexity metrics of Gr"obner bases for binomial ideals, highlighting the challenges and potential of machine learning in algebraic computations.

Contribution

It introduces a large dataset and demonstrates neural networks can effectively predict Gr"obner basis metrics, outperforming traditional regression methods.

Findings

Neural networks achieved an $r^2$ of 0.401 in predictions.

The dataset captures variability in Gr"obner complexity.

Predictions outperform naive and multiple regression models.

Abstract

We construct neural network regression models to predict key metrics of complexity for Gr\"obner bases of binomial ideals. This work illustrates why predictions with neural networks from Gr\"obner computations are not a straightforward process. Using two probabilistic models for random binomial ideals, we generate and make available a large data set that is able to capture sufficient variability in Gr\"obner complexity. We use this data to train neural networks and predict the cardinality of a reduced Gr\"obner basis and the maximum total degree of its elements. While the cardinality prediction problem is unlike classical problems tackled by machine learning, our simulations show that neural networks, providing performance statistics such as $r^2 = 0.401$, outperform naive guess or multiple regression models with $r^2 = 0.180$.