# Subgroup-based Rank-1 Lattice Quasi-Monte Carlo

arXiv: https://arxiv.org/abs/2011.06446 · HTML: https://mlsanity.com/paper/2011.06446/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2020.

## Built on by (most cited first, 3 of 3)

- [Spectraformer: A Unified Random Feature Framework for Transformer](https://mlsanity.com/paper/2405.15310/) (2024)
- [Neural Optimization Kernel: Towards Robust Deep Learning](https://mlsanity.com/paper/2106.06097/) (2021)
- [Subgroup Rank-1 Lattice for Practical High-dimensional Black-box Integral Approximation](https://mlsanity.com/paper/2609.35177/) (2026)

## Abstract (arXiv)

Quasi-Monte Carlo (QMC) is an essential tool for integral approximation, Bayesian inference, and sampling for simulation in science, etc. In the QMC area, the rank-1 lattice is important due to its simple operation, and nice properties for point set construction. However, the construction of the generating vector of the rank-1 lattice is usually time-consuming because of an exhaustive computer search. To address this issue, we propose a simple closed-form rank-1 lattice construction method based on group theory. Our method reduces the number of distinct pairwise distance values to generate a more regular lattice. We theoretically prove a lower and an upper bound of the minimum pairwise distance of any non-degenerate rank-1 lattice. Empirically, our methods can generate a near-optimal rank-1 lattice compared with the Korobov exhaustive search regarding the l1-norm and l2-norm minimum distance. Moreover, experimental results show that our method achieves superior approximation performance on benchmark integration test problems and kernel approximation problems.