ATD: Gaussian Fields: Graph Representations and Black-Box Optimization Algorithms

科学和工程中使用的数据量和模型的复杂性不断增加，需要在计算数学，统计学和机器学习的交叉点进行协作和跨学科研究。该项目将开发新的计算数学，目的是促进使用高斯场方法对大型和非结构化空间数据集进行统计分析。高斯场是统计学和机器学习中的标准模型，但将其应用于大型数据集是非常困难的。研究人员将通过数学推理和实际例子表明，一个标准的高斯场族可以用图形精确地近似，这样统计分析就可以有利地扩展到大数据集。因此，通过高斯场的声音统计建模的好处是有可能在当前感兴趣的大数据制度。此外，研究人员将探索新的计算效率高的方法，使一个联合收割机高度复杂的模型与数据相结合。该项目的一个核心部分将是培训芝加哥大学计算和应用数学（CAM）项目的研究生。为此，调查员将i）介绍CAM和统计课程中当前研究兴趣的不确定性量化和空间统计主题，无论是在硕士和博士水平; ii）担任CAM学生的博士和硕士论文顾问; ㈢帮助组织CAM座谈会和CAM学生座谈会，使学生能够向空间统计方面的主要专家学习并与他们互动，贝叶斯逆问题和基于图形的学习; iv）通过会议和研讨会演示来传播已完成的工作。该项目有两个研究重点，其共同主题是推动高斯场方法的使用。首先，研究者将开发和分析高斯场的图形表示，用于离散和非结构化空间数据集的统计分析。其次，研究者将设计和数值探索新的黑箱，衍生物自由优化计划，联合收割机贝叶斯优化与集合卡尔曼方法相结合。所使用的图形表示源于高斯场的随机偏微分方程（SPDE）方法，这是过去十年空间统计学的突破之一。SPDE方法的主要思想是将高斯场定义为SPDE的解，并使用有限元表示该解，这一观点激发了许多建模和计算发展。研究人员将介绍和探索图形表示，通过严格的分析和数值示例显示，它们提供了一种将Matérn模型推广到非结构化数据集的自然方法。研究人员还将证明，图形表示无缝统一了空间统计，基于图形的机器学习和贝叶斯逆问题中的高斯场方法，并将在这三个社区中转移几种具体的计算方法和建模思想。第二个研究重点将关注新的黑箱，衍生自由优化计划的发展。研究人员将对现有方法进行彻底的数值比较，并将探索新的方法。该项目的具体目标包括：㈠为大型和非结构化离散地理空间数据集引入基于图形的协方差模型，超越欧几里德设置; ㈡提出空间统计中一系列广泛模型的图形表示，为现有有限元和有限差分表示提供另一种方法; iii）提出高斯场的图形表示的理论基础，并研究它们在特定应用中的使用;以及iv）开发和数值测试新的黑盒优化方案，该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。

The increasing amount of data and complexity of models used in science and engineering calls for collaborative and interdisciplinary research at the intersection of computational mathematics, statistics and machine learning. This project will develop novel computational mathematics with the aim of facilitating the statistical analysis of large and unstructured spatial data-sets using Gaussian field methods. Gaussian fields are standard models in statistics and machine learning, but their application to large data sets is notoriously difficult. The investigator will show through mathematical reasoning and practical examples that a standard family of Gaussian fields can be accurately approximated using graphs in such a way that the statistical analysis scales favorably to large data sets. As a result, the benefit of sound statistical modeling through Gaussian fields is made possible in large data regimes of current interest. In addition, the investigator will explore new computationally efficient methods that allow one to combine highly complex models with data. A central part of the project will be the training of graduate students in the Computational and Applied Mathematics (CAM) program at the University of Chicago. To that end, the investigator will i) introduce topics of current research interest in uncertainty quantification and spatial statistics in CAM and Statistics courses, both at the Master’s and PhD levels; ii) serve as PhD and Master's thesis adviser of CAM students; iii) help organize the CAM colloquium and the CAM student colloquium, allowing students to learn from, and interact with, leading experts in spatial statistics, Bayesian inverse problems and graph-based learning; and iv) disseminate the accomplished work through conference and seminar presentations.This project has two research thrusts that share a common theme of pushing forward the use of Gaussian field methods. First, the investigator will develop and analyze graph representations of Gaussian fields for the statistical analysis of discrete and unstructured spatial datasets. Second, the investigator will design and numerically explore novel black-box, derivative free optimization schemes that combine Bayesian optimization with ensemble Kalman methods. The graph representations to be used stem from the stochastic partial differential equation (SPDE) approach to Gaussian fields, one of the breakthroughs in spatial statistics in the last decade. The main idea of the SPDE approach is to define Gaussian fields as the solution to an SPDE and represent the solution using finite elements, a perspective that has inspired many modeling and computational developments. The investigator will introduce and explore graph representations, showing through rigorous analysis and numerical examples that they provide a natural way to generalize the Matérn model to unstructured datasets. The investigator will also demonstrate that graph representations seamlessly unify Gaussian field methods in spatial statistics, graph-based machine learning and Bayesian inverse problems, and will transfer several concrete computational methods and modeling ideas across these three communities. The second research thrust will concern the development of new black-box, derivative free optimization schemes. The investigator will conduct a thorough numerical comparison of existing methods, and will explore new ones. Specific objectives of this project include i) to introduce graph-based covariance models for large and unstructured discrete geospatial datasets, beyond Euclidean settings; ii) to suggest graph representations of a wide family of models in spatial statistics, providing an alternative approach to existing finite element and finite difference representations; iii) to set forward the theoretical foundations of graph representations of Gaussian fields and investigate their use in specific applications; and iv) to develop and numerically test new black-box optimization schemes, exploring their use in a variety of applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.