Hilbert Series, Machine Learning, and Applications to Physics

TL;DR

This paper demonstrates that simple machine learning models can accurately predict geometric properties from Hilbert series, aiding in mathematical and physical applications.

Contribution

It introduces machine learning approaches to predict and classify properties of geometric objects from Hilbert series, showing high accuracy and ease of generating synthetic data.

Findings

Regressors predict embedding weights with ~1 MAE

Classifiers predict dimension and Gorenstein index with >90% accuracy

Random forests distinguish complete intersections with >95% accuracy

Abstract

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ${\sim}1$ mean absolute error, whilst classifiers predict dimension and Gorenstein index to $>90\%$ accuracy with ${\sim}0.5\%$ standard error. Binary random forest classifiers managed to distinguish whether the underlying HS describes a complete intersection with high accuracies exceeding $95\%$. Neural networks (NNs) exhibited success identifying HS from a Gorenstein ring to the same order of accuracy, whilst generation of 'fake' HS proved trivial for NNs to distinguish from those associated to the three-dimensional Fano varieties considered.