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

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.

Published in Proceedings of the National Academy of Sciences (PNAS): https://www.pnas.org/doi/10.1073/pnas.2409913121

References in corpus (12)

• Semi-Supervised Classification with Graph Convolutional Networks
• Neural Message Passing for Quantum Chemistry
• Graph Convolutional Neural Networks for Web-Scale Recommender Systems
• Spectral Networks and Locally Connected Networks on Graphs
• Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering
• ETA Prediction with Graph Neural Networks in Google Maps
• Graph Neural Networks in Particle Physics
• Calculation of Discrepancy Measures and Applications
• The construction of good lattice rules and polynomial lattice rules
• Constructing Low Star Discrepancy Point Sets with Genetic Algorithms
• Fully Parallel Hyperparameter Search: Reshaped Space-Filling
• Computing Star Discrepancies with Numerical Black-Box Optimization Algorithms