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Efficient computation of N -point correlation functions in D dimensions
Proceedings of the National Academy of Sciences of the United States of America ( IF 11.1 ) Pub Date : 2022-08-08 , DOI: 10.1073/pnas.2111366119
Oliver H. E. Philcox 1, 2 , Zachary Slepian 3, 4
Affiliation  

We present efficient algorithms for computing the N -point correlation functions (NPCFs) of random fields in arbitrary D -dimensional homogeneous and isotropic spaces. Such statistics appear throughout the physical sciences and provide a natural tool to describe stochastic processes. Typically, algorithms for computing the NPCF components have O ( n N ) complexity (for a dataset containing n particles); their application is thus computationally infeasible unless N is small. By projecting the statistic onto a suitably defined angular basis, we show that the estimators can be written in a separable form, with complexity O ( n 2 ) or O ( n g log n g ) if evaluated using a Fast Fourier Transform on a grid of size n g . Our decomposition is built upon the D -dimensional hyperspherical harmonics; these form a complete basis on the ( D 1 ) sphere and are intrinsically related to angular momentum operators. Concatenation of ( N 1 ) such harmonics gives states of definite combined angular momentum, forming a natural separable basis for the NPCF. As N and D grow, the number of basis components quickly becomes large, providing a practical limitation to this (and all other) approaches: However, the dimensionality is greatly reduced in the presence of symmetries; for example, isotropic correlation functions require only states of zero combined angular momentum. We provide a Julia package implementing our estimators and show how they can be applied to a variety of scenarios within cosmology and fluid dynamics. The efficiency of such estimators will allow higher-order correlators to become a standard tool in the analysis of random fields.

中文翻译:

D维N点相关函数的高效计算

我们提出了用于计算的有效算法- 任意随机场的点相关函数 (NPCFs)维齐次和各向同性空间。此类统计数据出现在整个物理科学中,并提供了描述随机过程的自然工具。通常,用于计算 NPCF 组件的算法具有 ( n ) 复杂性(对于包含n粒子); 因此,它们的应用在计算上是不可行的,除非是小。通过将统计量投影到适当定义的角度基础上,我们表明可以将估计量写成可分离的形式,具有复杂性 ( n 2个 ) 或者 ( n G 日志 n G ) 如果在大小为的网格上使用快速傅立叶变换进行评估 n G . 我们的分解建立在-维超球面谐波;这些构成了完整的基础 ( 1个 ) 球体并且与角动量算子有着内在的联系。串联的 ( 1个 ) 这种谐波给出了确定的组合角动量状态,形成了 NPCF 的自然可分离基础。作为随着增长,基础组件的数量迅速变大,为这种(以及所有其他)方法提供了实际限制:但是,在存在对称性的情况下,维数会大大降低;例如,各向同性相关函数只需要组合角动量为零的状态。我们提供了一个 Julia 包来实现我们的估计器,并展示了如何将它们应用于宇宙学和流体动力学中的各种场景。这种估计器的效率将使高阶相关器成为分析随机场的标准工具。
更新日期:2022-08-08
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