用点过程替代随机采样，让积分估计更准更快。
On two ways to use determinantal point processes for Monte Carlo integration
- 用确定性点过程代替独立采样，减少方差
- 两种方法分别达到O(N^-(1+1/d))和1/N的收敛速度
- 适用于需要高精度积分的数学与计算领域
标准蒙特卡洛估计器 $\widehat{I}_N^{\mathrm{MC}}$ 依赖于从 $ω$ 中独立采样，方差为 $1/N$ 量级。若用确定性点过程（DPP）替代采样，该过程具有排斥性，可使估计器保持一致，其方差速率取决于 DPP 如何适配函数 $f$ 和测度 $ω$。本文分析了两种已有基于 DPP 的估计器：一种由 Bardenet & Hardy（2020）提出，对光滑 $f$ 可达 $\mathcal{O}(N^{-(1+1/d)})$ 收敛率，但使用固定 DPP；另一种由 Ermakov & Zolotukhin（1960）提出，无偏且收敛率为 $1/N$，但 DPP 针对 $f$ 定制。本文重新审视这些方法，将其推广至连续情形，并给出相应的采样算法。
原文摘要 · Abstract (English)
The standard Monte Carlo estimator $\widehat{I}_N^{\mathrm{MC}}$ of $\int fdω$ relies on independent samples from $ω$ and has variance of order $1/N$. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to $f$ and $ω$. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of $\mathcal{O}(N^{-(1+1/d)})$ for smooth $f$, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order $1/N$, like Monte Carlo, but its DPP is tailored to $f$. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.
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