Predicting the Integer Decomposition Property via Machine Learning

TL;DR

This paper explores using neural networks to predict the integer decomposition property of lattice simplices by analyzing the distribution of Hilbert basis elements, enabling efficient identification of IDP simplices in large datasets.

Contribution

It introduces a neural network approach to predict algebraic properties of lattice simplices, specifically the IDP, from geometric data, which is a novel application in this area.

Findings

Neural networks can effectively predict the IDP in lattice simplices.

The method scales to large datasets for IDP detection.

The approach provides a new tool for algebraic and geometric analysis.

Abstract

In this paper we investigate the ability of a neural network to approximate algebraic properties associated to lattice simplices. In particular we attempt to predict the distribution of Hilbert basis elements in the fundamental parallelepiped, from which we detect the integer decomposition property (IDP). We give a gentle introduction to neural networks and discuss the results of this prediction method when scanning very large test sets for examples of IDP simplices.