ID: 2103.13436

Hilbert Series, Machine Learning, and Applications to Physics

March 24, 2021

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Jiakang Bao, Yang-Hui He, Edward Hirst, Johannes Hofscheier, Alexander Kasprzyk, Suvajit Majumder
High Energy Physics - Theory
Mathematics
Algebraic Geometry

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.

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