Predicting the cardinality and maximum degree of a reduced Gröbner basis
This work addresses a specific complexity prediction problem in computational algebra, but it is incremental as it applies existing neural network methods to a new domain with limited performance gains.
The paper tackled the problem of predicting the cardinality and maximum degree of reduced Gröbner bases for binomial ideals using neural networks, achieving an r² of 0.401 which outperformed naive guesses and multiple regression models with r² of 0.180.
We construct neural network regression models to predict key metrics of complexity for Gröbner bases of binomial ideals. This work illustrates why predictions with neural networks from Gröbner 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 Gröbner complexity. We use this data to train neural networks and predict the cardinality of a reduced Gröbner 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 $r^2 = 0.180$.