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

ATD：高斯场：图表示和黑盒优化算法

基本信息

批准号：
2027056
负责人：
Daniel Sanz-Alonso
金额：
$ 18.11万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2027056&HistoricalAwards=false
关键词：
ATD Gaussian Fields Graph Representations

项目摘要

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.

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领域进行协作和跨学科研究，该项目将开发新型计算数学，旨在促进大规模和大规模的统计分析。使用高斯场方法的非结构化空间数据集是统计学和机器学习中的标准模型，但众所周知，它们在大型数据集上的应用非常困难，研究人员将通过数学推理和实际示例证明标准高斯场族可以做到这一点。准确使用图形进行近似，使统计分析能够有利地扩展到大型数据集。因此，在当前感兴趣的大型数据体系中，通过高斯场进行健全的统计建模的好处成为可能。此外，研究人员将探索新的方法。该项目的核心部分是对芝加哥大学计算与应用数学（CAM）项目的研究生进行培训。 i) 介绍主题当前对 CAM 和统计学课程中的不确定性量化和空间统计的研究兴趣，包括硕士和博士级别；ii) 担任 CAM 学生的博士和硕士论文导师；iii) 帮助组织 CAM 学术讨论会和 CAM 学生学术讨论会学生向空间统计、贝叶斯反问题和基于图形的学习领域的领先专家学习并互动；iv) 通过会议和研讨会演示传播已完成的工作。该项目有两个研究主旨的共同主题是推动高斯场方法的使用，首先，研究人员将开发和分析高斯场的图形表示，以进行离散和非结构化空间数据集的统计分析；其次，研究人员将设计和数值探索。新颖的黑盒、无导数优化方案，将贝叶斯优化与集成卡尔曼方法相结合。所使用的图形表示源于高斯场的随机偏微分方程（SPDE）方法，这是空间领域的突破之一。 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，这一观点启发了许多建模和计算的发展。探索图形表示，通过严格的分析和数值示例表明它们提供了一种将 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地统一了空间统计、基于图形的机器学习和计算中的高斯场方法。贝叶斯逆问题，并将在这三个领域转移几种具体的计算方法和建模思想，研究人员将对现有方法进行彻底的数值比较。该项目的具体目标包括 i) 为大型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，超越欧几里得设置；ii) 提出空间统计中广泛模型系列的图形表示，提供另一种选择现有有限元和有限差分表示的方法；iii）提出高斯场图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，探索它们的使用该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。