论文标题
黑桃：通过差异估计的顺序聚类粒子歼灭
SPADE: Sequential-clustering Particle Annihilation via Discrepancy Estimation
论文作者
论文摘要
对于经验签名的度量，$μ= \ frac {1} {n} \ left（\ sum_ {i = 1}^pΔ__{x_i} - \ sum_ {i = 1}^mδ__{y__i} $ \ {x_i \} _ {i = 1}^p $和$ \ {y_i \ \} _ {i = 1}^m $同时产生另一个经验签名的度量$ν$，因此$ \ int fdν$近似于$ \ int f d uncections of Coptable tocception test ockable test offectable test可接受的函数，以便适用于适当的测试功能。携带相反重量的颗粒的这种灭绝已被广泛用于减轻粒子模拟中的数值符号问题。在本文中，我们基于聚类和匹配的杂种在高维欧几里得空间中提出了PA的算法，并通过差异估计（Spade）称为顺序聚类粒子歼灭。它由两个步骤组成：通过控制粒子的数量理论差异的自适应聚类，以及每个集群中正和负粒子之间的独立随机匹配。 koksma-hlawka不等式和非质子随机误差范围均受到浓度不平等的界限。一个因素可以衡量点分布的不规则性并反映其离散的性质。另一个依赖于测试功能的变化，并受到连续性的影响。只有后者隐含地取决于维度$ d $，这意味着Spade可以免疫各种测试功能的尺寸诅咒。数值实验最多$ d = 1080 $验证我们的理论发现。
For an empirical signed measure $μ= \frac{1}{N} \left(\sum_{i=1}^P δ_{x_i} - \sum_{i=1}^M δ_{y_i}\right)$, particle annihilation (PA) removes $N_A$ particles from both $\{x_i\}_{i=1}^P$ and $\{y_i\}_{i=1}^M$ simultaneously, yielding another empirical signed measure $ν$ such that $\int f d ν$ approximates to $\int f d μ$ within an acceptable accuracy for suitable test functions $f$. Such annihilation of particles carrying opposite importance weights has been extensively utilized for alleviating the numerical sign problem in particle simulations. In this paper, we propose an algorithm for PA in high-dimensional Euclidean space based on hybrid of clustering and matching, dubbed the Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE). It consists of two steps: Adaptive clustering of particles via controlling their number-theoretic discrepancies, and independent random matching among positive and negative particles in each cluster. Both deterministic error bounds by the Koksma-Hlawka inequality and non-asymptotic random error bounds by concentration inequalities are proved to be affected by two factors. One factor measures the irregularity of point distributions and reflects their discrete nature. The other relies on the variation of test function and is influenced by the continuity. Only the latter implicitly depends on dimensionality $d$, implying that SPADE can be immune to the curse of dimensionality for a wide class of test functions. Numerical experiments up to $d=1080$ validate our theoretical discoveries.
