On the reconstruction of kinematic distributions computed with Monte Carlo methods using orthogonal basis functions

TL;DR

The paper addresses the challenge of reconstructing one-dimensional kinematic distributions from high-dimensional Monte Carlo integrations without relying on histograms. It develops a moments-based framework using an orthonormal basis, notably Legendre moments, with coefficients c_k = \langle f,e_k\rangle computed via MC, and provides a truncation criterion to balance Monte Carlo errors against approximation error. A key contribution is the proposal of a rectified, optimized basis built from a known approximate distribution f_0(x), which improves convergence and tail behavior in perturbative QCD calculations, demonstrated in Higgs production via weak-boson fusion. The approach yields smooth, differentiable distributions that are less sensitive to bin-to-bin fluctuations from local subtraction schemes, and the results generalize to higher dimensions, offering a practical alternative or complement to conventional histograms in MC integrators.

Abstract

Reconstruction of one-dimensional kinematic distributions from calculations based on high-dimensional Monte-Carlo integration is a standard problem in high-energy physics. Traditionally, this is done by collecting randomly-generated events in histograms. In this article, we explore an alternative approach, whose main idea is to approximate the target distribution by a weighted sum of orthogonal basis functions whose coefficients are calculated using the Monte-Carlo integration. This method has the advantage of directly yielding smooth approximations to target distributions. Furthermore, in the context of high-order perturbative calculations with local subtractions, it eliminates the so-called bin-to-bin fluctuations, which often severely affect the quality of conventional histograms. We also demonstrate that the availability of a high-quality approximation to the target distribution, for example the leading-order result in the perturbative expansion, can be exploited to construct an optimized orthonormal basis. We compare the performance of this method to conventional histograms in both toy-model and real Monte-Carlo settings, applying it to Higgs boson production in weak boson fusion as an example.

Figures (9)

• Figure 1: Legendre moments $\langle f,e_k\rangle$ of the normal distribution $f= \mathcal{N}(0,1)$ (see Eq. (\ref{['eq22']}) for the definition) on the interval $[-1,2]$. The blue "Accurate" crosses are the true Legendre moments, calculated with high precision. The black data points are the moments calculated in a toy Monte-Carlo with $n=10^5$ samples. The orange dashed line is the estimate $\langle f,e_k\rangle\sim cr^k$ of the typical magnitude of the Legendre moments, obtained from the Monte-Carlo data (black points). The vertical gray dashed line marks the truncation point, determined according to the prescription described in Sec. \ref{['sec:truncation']}. The distribution itself and its reconstruction from the retained moments are shown in Figure \ref{['fig:toy']}.
• Figure 2: Normal distribution $(2\pi)^{-1/2}\exp(-x^2/2)$ reconstructed from $n$ samples using a histogram or Legendre moments. All plots, except for the bottom-left one, compare the two approximations on the interval $[-1,2]$. The bottom-left plot shows the approximation on the interval $[-1.5,5]$. The plots in the top row differ in the number of Monte-Carlo samples $n$. The histograms in the right column differ in the bin width, while the Legendre approximation is the same. The number in parenthesis indicates the number of retained terms in the Legendre series, which is determined from the estimated Monte-Carlo uncertainties according to the truncation prescription described in Sec. \ref{['sec:truncation']}. The bottom panels show the relative deviation of the reconstructed distributions from the true distribution. For each plot, the histogram and Legendre approximation are reconstructed using the same Monte-Carlo samples.
• Figure 3: Examples of distributions that are more difficult to approximate using Legendre moments. The distribution on the left is a mixture of two gaussian peaks. The distribution on the right, defined in Eq. \ref{['eq:smoothstep']} has an infinite derivative at $x=0$. The number in parenthesis indicates the number of retained terms in the Legendre series, which is determined from the estimated Monte-Carlo uncertainties according to the truncation prescription described in Sec. \ref{['sec:truncation']}. The bottom panels show the relative deviation of the reconstructed distributions from the true distribution.
• Figure 4: Distribution of transverse momentum $p_{\perp,H}$ of Higgs boson produced in WBF, reconstructed using histograms (left) or Legendre moments (right). The blue, orange, and black lines correspond to the leading-, next-to-leading, and next-to-next-to-leading order result. The bottom panels show relative deviations from the leading-order one. For Legendre moments the number in brackets indicates the number of moments retained after the truncation procedure described in Section \ref{['sec:truncation']}. The uncertainty bands account for Monte-Carlo integration uncertainty, and, for Legendre moments, the truncation uncertainty.
• Figure 5: Distribution of rapidity $y_H$ of Higgs boson produced in WBF, reconstructed using histograms (left) or Legendre moments (right). The blue, orange, and black lines correspond to the leading-, next-to-leading, and next-to-next-to-leading order result. The bottom panels show relative deviations from the leading-order one. For Legendre moments the number in brackets indicates the number of moments retained after the truncation procedure described in Section \ref{['sec:truncation']}. The uncertainty bands account for Monte-Carlo integration uncertainty, and, for Legendre moments, the truncation uncertainty.