Fast n-point correlation functions and three-point lensing application

TL;DR

The paper introduces a fast, tree-based algorithm for computing two-point, three-point, and general n-point correlation functions using a balanced monopole binary tree, achieving roughly O(N log N) to O(N theta_c^-4 log terms) scaling in 2D. It details mainstream and accuracy-driven subdivision strategies, and provides explicit node-to-node computations for both scalar and spin-2 fields, including weak gravitational lensing applications. The method generalizes to n-PCF in k dimensions, with a configurable accuracy parameter theta_c, and demonstrates favorable speed, accuracy, and memory characteristics compared with brute force and FFT-based approaches. This framework enables full 3PCF analyses on catalogs with up to around a million galaxies and is poised for application to upcoming large-scale surveys in lensing and CMB studies.

Abstract

We present a new algorithm to rapidly compute the two-point (2PCF), three-point (3PCF) and n-point (n-PCF) correlation functions in roughly O(N log N) time for N particles, instead of O(N^n) as required by brute force approaches. The algorithm enables an estimate of the full 3PCF for as many as 10^6 galaxies. This technique exploits node-to-node correlations of a recursive bisectional binary tree. A balanced tree construction minimizes the depth of the tree and the worst case error at each node. The algorithm presented in this paper can be applied to problems with arbitrary geometry. We describe the detailed implementation to compute the two point function and all eight components of the 3PCF for a two-component field, with attention to shear fields generated by gravitational lensing. We also generalize the algorithm to compute the n-point correlation function for a scalar field in k dimensions where n and k are arbitrary positive integers.

Figures (5)

• Figure 1: For a set of seven points, we first find the weighted average position; in this plot, it is simply the centre of mass, labeled by A7, because we choose equal weight for each point. We subdivide the root node by cutting through a projected median point perpendicular to the line connecting the weighted centre of mass and the furthest particle from it, that is, the dashed line emitting from A7. We then recursively subdivide each of the daughter nodes in the same manner. The character-number labels in this figure indicate the placement of the nodes, whose centres of mass coincide with the small circles, in the tree hierarchy, as shown in figure \ref{['fig:buildtree2']}.
• Figure 3: For $1000$ particles, we plot the fractional error of the 3PCF for a scalar field computed using the new algorithm compared to that using the brute force approach. Recall that the 3PCF is stored on a 3D grid. We first smooth over the neighbouring grids, then compute the fractional error $\Delta=\sqrt{\frac{\sum_{i,j,k}(\xi_{\theta_c}-\xi_{\ast})^2}{ \sum_{i,j,k}\xi{\ast}^2}}$ where $\xi_{\ast}$ is the 3PCF computed by direct summation over all triplets of particles. We see that the truncation error scales as $\theta_c^2$ which agrees with our estimate.
• Figure 4: A plot of the total raw 3PCF computing time for the shear, $\kappa$ and the weight, against the number of particles. While the raw correlation functions for $\kappa$ and for the weight are scalar, the shear raw correlation function consists of 8 components. The different lines correspond to different $\theta_c$. The slopes given in the plot are measured at a fixed computing time of 1000 seconds. One expects the slope to approach $1$ as $\theta_c^2 N \rightarrow \infty$. The plot also shows the average tree building time over the different $\theta_c$'s.
• Figure 5: A plot of the scaling index versus $\theta_c$ for different values of $N$. Here, the scaling index is defined to be the log-log slope of the computing time versus $N$. On each curve plotted here, the scaling indices are obtained by measuring the slopes in figure \ref{['fig:speed_n']} at a fixed $N$. However, curves for $\theta_c=0.02$ and $0.05$ are not shown in figure \ref{['fig:speed_n']} to avoid crowdedness.
• Figure 6: A plot of the total raw 3PCF computing time for the shear, $\kappa$ and the weight against $\theta_c$ for $N=5000$. While the raw correlation functions for $\kappa$ and for the weight are scalar, the shear raw correlation function consists of 8 components. A line with slope$=-4$, which is expected to be the limit for large $\theta_c$, is also plotted for the purpose of comparison.