Linear statistics of point processes yield Monte Carlo estimators of integrals. While the simplest approach relies on a homogeneous Poisson point process, more regularly spread point processes, such as scrambled low-discrepancy sequences or determinantal point processes, can yield Monte Carlo estimators with fast-decaying mean square error. Following the intuition that more regular configurations result in lower integration error, we introduce the repulsion operator, which reduces clustering by slightly pushing the points of a configuration away from each other. Our main theoretical result is that applying the repulsion operator to a homogeneous Poisson point process yields an unbiased Monte Carlo estimator with lower variance than under the original point process. On the computational side, the evaluation of our estimator is only quadratic in the number of integrand evaluations and can be easily parallelized without any communication across tasks. We illustrate our variance reduction result with numerical experiments and compare it to popular Monte Carlo methods. Finally, we numerically investigate a few open questions on the repulsion operator. In particular, the experiments suggest that the variance reduction also holds when the operator is applied to other motion-invariant point processes.

翻译：点过程的线性统计量可生成积分的蒙特卡罗估计量。尽管最简方法依赖于齐次泊松点过程，但更规则分布的点过程（如加扰低差异序列或行列式点过程）能产生均方误差快速衰减的蒙特卡罗估计量。基于“更规则配置导致更低积分误差”的直觉，我们引入排斥算子——通过轻微推开配置中的点来减少聚类效应。主要理论结果是：将排斥算子应用于齐次泊松点过程，所生成的无偏蒙特卡罗估计量方差低于原始点过程。计算方面，该估计量的评估复杂度仅为被积函数评估次数的二次方，且可轻松并行化而无需任务间通信。我们通过数值实验验证方差缩减结果，并与主流蒙特卡罗方法进行比较。最后，针对排斥算子的若干开放性问题进行数值探究。实验表明，当算子应用于其他运动不变点过程时，方差缩减特性依然成立。