论文标题
基于亚组的等级-1晶格准蒙特卡洛
Subgroup-based Rank-1 Lattice Quasi-Monte Carlo
论文作者
论文摘要
准蒙特卡洛（QMC）是积分近似，贝叶斯推断和科学仿真等取样的必不可少的工具。在QMC区域中，由于其简单的操作以及对点集构建的良好属性，等级-1晶格很重要。但是，由于详尽的计算机搜索，等级-1晶格的生成向量的构建通常很耗时。为了解决这个问题，我们提出了一种基于群体理论的简单封闭形式的等级-1晶格构建方法。我们的方法减少了不同的成对距离值的数量，以生成更常规的晶格。从理论上讲，我们证明了任何非分类级别-1晶格的最小成对距离的较低和上限。从经验上讲，与Korobov详尽的搜索有关$ L_1 $ -NORM和$ L_2 $ -NORM最小距离的详尽搜索，我们的方法可以产生近乎最佳的等级-1晶格。此外，实验结果表明，我们的方法在基准集成测试问题和内核近似问题上实现了出色的近似性能。
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.
