Subgroup-based Rank-1 Lattice Quasi-Monte Carlo

TL;DR

This paper introduces a novel, group theory-based closed-form construction method for rank-1 lattices in Quasi-Monte Carlo, reducing computational effort and improving point set regularity for better integration accuracy.

Contribution

It presents a new simple construction method for rank-1 lattices that avoids exhaustive search, with theoretical bounds and empirical evidence of near-optimality and improved performance.

Findings

Reduces construction time compared to exhaustive search.

Achieves near-optimal minimum pairwise distances.

Improves integration and kernel approximation accuracy.

Abstract

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 $l_1$-norm and $l_2$-norm minimum distance. Moreover, experimental results show that our method achieves superior approximation performance on benchmark integration test problems and kernel approximation problems.