Message-Passing Monte Carlo: Generating low-discrepancy point sets via Graph Neural Networks

TL;DR

This paper introduces Message-Passing Monte Carlo, a novel machine learning method using Graph Neural Networks to generate low-discrepancy point sets, improving uniformity and performance in various scientific and engineering applications.

Contribution

It presents the first ML approach to generate low-discrepancy points with a GNN-based model, extending to higher dimensions and demonstrating superior performance over previous methods.

Findings

MPMC points achieve state-of-the-art discrepancy performance.

The method is near-optimal for low dimensions and small point sets.

Code is publicly available for reproducibility.

Abstract

Discrepancy is a well-known measure for the irregularity of the distribution of a point set. Point sets with small discrepancy are called low-discrepancy and are known to efficiently fill the space in a uniform manner. Low-discrepancy points play a central role in many problems in science and engineering, including numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. In this work, we present the first machine learning approach to generate a new class of low-discrepancy point sets named Message-Passing Monte Carlo (MPMC) points. Motivated by the geometric nature of generating low-discrepancy point sets, we leverage tools from Geometric Deep Learning and base our model on Graph Neural Networks. We further provide an extension of our framework to higher dimensions, which flexibly allows the generation of custom-made points that emphasize the uniformity in specific dimensions that are primarily important for the particular problem at hand. Finally, we demonstrate that our proposed model achieves state-of-the-art performance superior to previous methods by a significant margin. In fact, MPMC points are empirically shown to be either optimal or near-optimal with respect to the discrepancy for low dimension and small number of points, i.e., for which the optimal discrepancy can be determined. Code for generating MPMC points can be found at https://github.com/tk-rusch/MPMC.