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 property for point set construction. However, the construction of the generating vector of the rank-1 lattice is usually time-consuming through 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 near-optimal rank-1 lattice compared with 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 the benchmark integration test problems and the kernel approximation problems.

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 property for point set construction. However, the construction of the generating vector of the rank-1 lattice is usually time-consuming through 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.

Weaknesses: My main concerns are whether this work makes an impactful contribution to the type of problems of interest to the NeurIPS community. The problem of high-dimensional integration is of course of vital importance in many areas of machine learning, appearing centrally for example in Bayesian inference/model selection, graphical models, and the training of latent variable generative models, and many of us would welcome an addition to the toolkit of dealing with such beasts. Unfortunately, this paper makes only minimal effort to motivate the relevance of the proposed QMC construction to these settings. An application to GAN/VAEs does briefly appear in the supplementary, but with quite cursory quantification of performance; showing sharper generated images is not consistent with the rigorous aims and tone of the paper. For a NeurIPS audience, I consider it essential to include a comparison against established sampling algorithms such as Sequential Monte Carlo.

This paper is very much borderline and sparked an extensive discussion among reviewers. On the positive side, this work presents a simple closed form generation rule for rank-1 lattice in QMC, which previously required exhaustive search. The method is novel and solid, with promising empirical results. On the negative sides, all reviewers have concerns with 1) the lack of comparison to methods in ML community, and the fitness of the venue (however, the target problem of integration approximation is of high importance in Neurips as well); 2) some limited clarity/questions on empirical methodology; and 3) some writing quality /typo issues. The authors did an excellent job in rebuttal, including providing some initial results on a toy restricted Boltzmann model.

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 property for point set construction. However, the construction of the generating vector of the rank-1 lattice is usually time-consuming through 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.