Deterministic Sampling through Energy Minimization

该项目旨在开发用于统计抽样/统计观测的最佳确定性方法，与常用的随机抽样方法（如Monte Carlo（MC）和Markov Chain Monte Carlo（MCMC））相比。MC/MCMC方法彻底改变了统计学，使统计学家能够建模和解决使用传统技术难以解决的复杂和高维问题。这些方法的一个缺点是，由于随机采样中固有的缓慢收敛速度，需要非常多的观测或数据样本。当采样昂贵时，这成为一个问题。本项目正在研究的确定性方法试图通过更智能地采样点来克服这一问题，从而可以用更少的确定性样本获得随机样本提供的相同信息。这可以显著降低采样和后续计算的成本。正在开发的方法在许多领域都有应用，例如不确定性量化，计算机实验和机器学习。该项目旨在通过最小化某些能量来提供确定性样本。我们的目标是使用精心开发的优化技术，以减少昂贵的概率分布评估的数量，从而降低整体计算成本。此外，确定性样本为分布提供了更好的代表性点集，这可以进一步降低涉及积分的后续计算的成本。与现有的主要针对均匀超立方体抽样的拟蒙特卡罗方法相比，本文所研究的方法具有更好的通用性，可以直接从任意概率分布中进行抽样。将研究确定性抽样的两种方法。第一种方法，称为最小能量设计，当概率密度的计算成本很高时很有用。第二种方法，称为支持点，当被积函数很昂贵但从概率密度采样很容易时很有用。最小能量设计具有一个重要性质：其经验分布渐近收敛于目标分布。这是一个属性不共享的一些竞争的代表点集在文献中，如主点。另一方面，通过最小化用于拟合优度测试的能量距离来获得支持点。从这个角度来看，支持点可以被看作是最佳压缩连续概率分布的点集。该项目的重点是为这些能量函数开发高效的优化方法，使用尽可能少的函数评估，并改善点集的分布特性，使它们可以用于MC/MCMC方法在计算上不可行的问题。

This project aims at developing optimal deterministic methods for statistical sampling / statistical observations, in contrast to commonly-used random sampling methods such as Monte Carlo (MC) and Markov Chain Monte Carlo (MCMC). The MC/MCMC methods have revolutionized statistics, allowing statisticians to model and solve complex and high-dimensional problems that would have been intractable using conventional techniques. One drawback of these methods is that very many observations or data samples are needed due to the slow convergence rate inherent in random sampling. This becomes an issue when the sampling is expensive. The deterministic method under study in this project attempts to overcome this problem by sampling points more intelligently, so that the same information provided by a random sample can be obtained with fewer deterministic samples. This can significantly cut down the cost of sampling and subsequent computations. The method under development has applications in many fields, such as uncertainty quantification, computer experiments, and machine learning.The project aims to provide deterministic samples obtained through the minimization of certain energies. The goal is to use carefully developed optimization techniques to reduce the number of expensive evaluations of a probability distribution, thereby reducing the overall computational cost. Furthermore, the deterministic sample provides a much better representative set of points for the distribution, which can further reduce the cost of subsequent computations involving integrals. Compared to the existing Quasi-Monte Carlo methods, which are mostly developed for sampling from the uniform hypercube, the methods under study are much more general and can be used to directly sample from any probability distribution. Two methods for deterministic sampling will be investigated. The first method, known as minimum energy designs, is useful when the probability density is expensive to evaluate. The second method, known as support points, is useful when the integrand is expensive but sampling from the probability density is easy. The minimum energy design possesses an important property: its empirical distribution asymptotically converges to the target distribution. This is a property not shared by some of the competing representative point sets in the literature, such as principal points. On the other hand, support points are obtained by minimizing an energy distance which is used for goodness-of-fit testing. In this light, support points can be viewed as point sets that optimally compact a continuous probability distribution. The project focuses on developing efficient optimization methods for these energy functions using as few function evaluations as possible, and improving the distributional properties of the point sets so that they can be used in problems where MC/MCMC methods are computationally impracticable.