发表机构
University of California, Berkeley; Lawrence Berkeley National Laboratory（加州大学伯克利分校; 劳伦斯伯克利国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究用高斯过程解决含噪截断Hausdorff矩问题，以从矩中重构部分子分布函数，引入均值偏差校正和可学习相关长度核，在5个数据集上验证了稳健性。
AI 中文摘要
在这项工作中，我们研究了在从其矩恢复部分子分布函数（PDFs）的背景下，含噪截断Hausdorff矩问题。我们首先回顾Hausdorff矩问题的基础，并分析所提供的矩数据量与$f(x)$可重构能力之间的关系如何在不同类别的部分子分布函数中变化；然后我们转向使用以高斯过程（GPs）为中心的贝叶斯方法来解决矩问题。我们详细介绍了贝叶斯回归的标准方面，如协方差核选择、后验的解析表征、用蒙特卡洛方法进行超参数采样，以及验证指标。此外，我们引入了强大的新工具，例如一种校正后验分布中均值偏差的方法，以及一类简单的协方差核，其$x$相关长度在采样过程中被学习。最后，我们在来自价夸克、胶子和海夸克现象学π介子和核子PDF数据集的5个数据集上测试了我们的框架，这些数据集被选择以反映PDF行为的多样性。我们发现，在给定中等数量的矩的情况下，我们的GP框架能够稳健地重构PDFs。此外，本工作中开发的新工具解决了通过常规方法难以解决的持续性问题。所提出的框架可以容易地应用于通过格点量子色动力学（LQCD）获得的矩，以从理论第一性原理重构PDFs。
英文摘要
In this work, we study the noisy truncated Hausdorff moment problem in the context of recovering parton distribution functions (PDFs) from their moments. We start by reviewing basics of the Hausdorff moment problem and analyze how the relation between the amount of moment data provided and the ability for $f(x)$ to be reconstructed varies across different classes of parton distribution functions; we then turn to using Bayesian methods centered around Gaussian Processes (GPs) to solve the moment problems. Standard aspects of Bayesian regression such as covariance kernel choice, analytical characterization of the posterior, hyperparameter sampling with Monte Carlo methods, and validation metrics are detailed. In addition to this, powerful novel tools, such as a method to correct for mean bias in posterior distributions, and a simple class of covariance kernels whose $x$ correlation length is learned during sampling are introduced. Finally, we test our framework on 5 datasets from valence, gluon, and sea quark phenomenological pion and nucleon PDF datasets, selected to reflect diversity in PDF behavior. We find that our GP framework robustly reconstructs PDFs given a moderate number of moments. Additionally, the novel tools developed in this work resolve persistent issues that would be difficult to address via conventional means. The presented framework can be readily applied to moments obtained via lattice Quantum Chromodynamics (LQCD) to reconstruct PDFs from theoretical first principles.
Comments39 pages, 36 figures
