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

2023-02-10

AI Summary

This work addresses the problem of predicting the cardinality and maximum total degree of reduced Gröbner bases for binomial ideals—a core complexity measure in computational algebra. We propose the first neural network regression model specifically designed for Gröbner complexity prediction. Leveraging a large-scale, manually labeled dataset comprising two classes of highly variable random binomial ideals, we pioneer the application of deep learning to forecasting these critical symbolic computation metrics. The study identifies nontrivial challenges in aligning symbolic computation features with machine learning modeling requirements, and introduces a tailored data representation scheme and evaluation framework. Experimental results show that our model achieves an R² score of 0.401 for basis cardinality prediction—substantially outperforming both naive baselines and multivariate linear regression (R² = 0.180). We publicly release the first benchmark dataset for Gröbner complexity prediction, establishing a new paradigm for interdisciplinary research at the intersection of symbolic computation and machine learning.

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 $r^2 = 0.180$.

Problem

• Predicting cardinality of reduced Gröbner bases
• Estimating maximum degree in binomial ideal computations
• Assessing neural network performance on Gröbner complexity

Innovation

• Neural networks predict Gröbner basis cardinality
• Neural networks estimate maximum degree elements
• Generated probabilistic models dataset training variability