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Polytopes and Machine Learning

Jiakang Bao, Yang-Hui He, Edward Hirst, Johannes Hofscheier, Alexander Kasprzyk, Suvajit Majumder

TL;DR

This work focuses on 2d polygons and 3d polytopes with Pl\"ucker coordinates as input, which out-perform the usual vertex representation.

Abstract

We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d polygons and 3d polytopes with Plücker coordinates as input, which out-perform the usual vertex representation.

Polytopes and Machine Learning

TL;DR

This work focuses on 2d polygons and 3d polytopes with Pl\"ucker coordinates as input, which out-perform the usual vertex representation.

Abstract

We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d polygons and 3d polytopes with Plücker coordinates as input, which out-perform the usual vertex representation.

Paper Structure

This paper contains 29 sections, 3 theorems, 11 equations, 11 figures, 22 tables.

Table of Contents

  1. Introduction
  2. Polytope Preliminaries
  3. Plücker Coordinates
  4. Remark:
  5. Polytope Datasets
  6. Polygons in dimension 2
  7. Polytopes in dimension 3
  8. Machine Learning
  9. Methodology
  10. MLP for 2d polygons
  11. MLP for 3d polytopes
  12. CNN for reflexivity
  13. Random forest for reflexivity
  14. Polygons
  15. Predicting Polygon Properties
  16. ...and 14 more sections

Key Result

Theorem 2.1

In any dimension $d$, there is a finite number of canonical Fano polytopes, as well as reflexive polytopes, up to $\textup{GL}(d, \mathbb{Z})$, which acts on the vertices.

Figures (11)

  • Figure 2.1: The lattice polygons which are dual of each other.
  • Figure 3.1: Distributions of the polygon dataset's properties, split by the number of polygon's vertices, $n$. (a) shows the frequencies of polygons with each number of vertices in the dataset. (b-e) show the distributions over: volume, dual volume, Gorenstein index, and codimension respectively.
  • Figure 3.2: (a) The distribution of the volumes for 800000 random samples. (b) The distribution of the dual volumes for 800000 random samples.
  • Figure 3.3: (a) The distribution of the data points corresponding to reflexive polytopes, in terms of the numbers of their vertices. (b) The distribution of the 42943 randomly chosen data points corresponding to non-reflexive polytopes, in terms of the numbers of their vertices.
  • Figure 4.1: The true values vs NN predictions for ML volume (a-d), and dual volume (e-h), over the range of number of vertices, $n$, considered. Plots show the $y=x$ line for ease of comparison; and the MAE values are also given for each plot in the respective subcaptions to 3 decimal places.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • Example 1
  • Theorem 2.2: Cayley-Menger
  • Definition 2.6
  • Definition 2.7
  • ...and 6 more