Deterministic Sampling through Energy Minimization

通过能量最小化进行确定性采样

• 批准号: 1712642
• 负责人: Roshan Joseph
• 金额: $ 20万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712642&HistoricalAwards=false
• 关键词: Deterministic; Sampling; through; Energy; Minimization

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.

与蒙特卡罗(MC)和马尔可夫链蒙特卡罗(MCMC)等常用的随机抽样方法不同，本项目旨在开发统计抽样/统计观测的最优确定性方法。MC/MCMC方法使统计学发生了革命性的变化，使统计学家能够模拟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，由于随机抽样固有的收敛速度慢，需要非常多的观测值或数据样本。当抽样费用很高时，这就成了一个问题。本项目中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，以便用更少的确定性样本获得由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，如不确定性量化、计算机实验和机器学习。该项目旨在提供通过最小化特定能量获得的确定性样本。其目标是使用精心开发的优化技术来减少对概率分布进行昂贵评估的次数，从而降低总体计算成本。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。与现有的拟蒙特卡罗方法相比，现有的拟蒙特卡罗方法大多是为从均匀超立方体中抽样而开发的，所研究的方法要通用得多，可以直接从任何概率分布中抽样。我们将研究两种确定性抽样方法。第一种方法被称为最小能量设计，当概率密度评估成本较高时，该方法很有用。第二种方法称为支撑点，当被积函数很昂贵，但从概率密度进行采样很容易时，这种方法很有用。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。这是文献中一些竞争的代表性点集所不具有的性质，例如主点。另一方面，支撑点通过最小化用于拟合优度测试的能量距离来获得。从这个意义上讲，支撑点可以被视为最佳地紧凑连续概率分布的点集。该项目致力于用尽可能少的函数求值来开发这些能量函数的高效优化方法，并改善点集的分布特性，以便它们可以用于MC/MCMC方法在计算上不可行的问题。

项目成果

Space-Filling Designs for Robustness Experiments

稳健性实验的空间填充设计

• DOI: 10.1080/00401706.2018.1451390
• 发表时间: 2018
• 期刊: Technometrics
• 影响因子: 2.5
• 作者: Joseph, V. Roshan;Gu, Li;Ba, Shan;Myers, William R.
• 通讯作者: Myers, William R.

Transformation and Additivity in Gaussian Processes

• DOI: 10.1080/00401706.2019.1665592
• 发表时间: 2020-10
• 期刊: Technometrics
• 影响因子: 2.5
• 作者: Li-Hsiang Lin;V. R. Joseph

Deterministic Sampling of Expensive Posteriors Using Minimum Energy Designs

• DOI: 10.1080/00401706.2018.1552203
• 发表时间: 2017-12
• 期刊: Technometrics
• 影响因子: 2.5
• 作者: V. R. Joseph;Dianpeng Wang;Li Gu;Shiji Lyu;Rui Tuo

Uncertainty quantification of machining simulations using an in situ emulator

• DOI: 10.1080/00224065.2018.1474689
• 发表时间: 2018-07
• 期刊: Journal of Quality Technology
• 影响因子: 2.5
• 作者: Evren Gul;V. R. Joseph;Huan Yan;S. Melkote

Designing computer experiments with multiple types of factors: The MaxPro approach

• DOI: 10.1080/00224065.2019.1611351
• 发表时间: 2020-10
• 期刊: Journal of Quality Technology
• 影响因子: 2.5
• 作者: V. R. Joseph;Evren Gul;Shan Ba