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

arXiv: https://arxiv.org/abs/2405.15059 · HTML: https://mlsanity.com/paper/2405.15059/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 77 references that are in the map.
- First posted on arXiv 23 May 2024.

## Builds on (most cited first, 19 of 19)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Graph Convolutional Neural Networks for Web-Scale Recommender Systems](https://mlsanity.com/paper/1806.01973/) (2018)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [ETA Prediction with Graph Neural Networks in Google Maps](https://mlsanity.com/paper/2108.11482/) (2021)
- [Enhancing accuracy of deep learning algorithms by training with low-discrepancy sequences](https://mlsanity.com/paper/2005.12564/) (2020)
- [Higher-order Quasi-Monte Carlo Training of Deep Neural Networks](https://mlsanity.com/paper/2009.02713/) (2020)
- [Fully Parallel Hyperparameter Search: Reshaped Space-Filling](https://mlsanity.com/paper/1910.08406/) (2019)
- [Heuristic Approaches to Obtain Low-Discrepancy Point Sets via Subset Selection](https://mlsanity.com/paper/2306.15276/) (2023)
- [Star Discrepancy Subset Selection: Problem Formulation and Efficient Approaches for Low Dimensions](https://mlsanity.com/paper/2101.07881/) (2021)
- [Computing Star Discrepancies with Numerical Black-Box Optimization Algorithms](https://mlsanity.com/paper/2306.16998/) (2023)
- [Constructing Optimal L∞ Star Discrepancy Sets](https://mlsanity.com/paper/2311.17463/) (2023)
- [Constructing Low Star Discrepancy Point Sets with Genetic Algorithms](https://mlsanity.com/paper/1304.1978/) (2013)

## Built on by (most cited first, 6 of 6)

- [Low Stein Discrepancy via Message-Passing Monte Carlo](https://mlsanity.com/paper/2503.21103/) (2025)
- [An Iterative Deep Ritz Method for Monotone Elliptic Problems](https://mlsanity.com/paper/2501.15186/) (2025)
- [Neural Low-Discrepancy Sequences](https://mlsanity.com/paper/2510.03745/) (2025)
- [Optimizing Kernel Discrepancies via Subset Selection](https://mlsanity.com/paper/2511.02706/) (2025)
- [LLM-Guided Evolutionary Program Synthesis for Quasi-Monte Carlo Design](https://mlsanity.com/paper/2510.03650/) (2025)
- [A Bayesian Approach to Low-Discrepancy Subset Selection](https://mlsanity.com/paper/2602.14607/) (2026)

## Abstract (arXiv)

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.