Hilbert series, machine learning, and applications to physics. (English) Zbl 1487.81035
Summary: 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.
MSC:
| 81P68 | Quantum computation |
| 68T05 | Learning and adaptive systems in artificial intelligence |
| 68T07 | Artificial neural networks and deep learning |
| 16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
| 81P73 | Computational stability and error-correcting codes for quantum computation and communication processing |
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