Parameter Estimation Theory and Algorithms under Latent Variable Models and Model Misspecification

潜变量模型已成为现代统计学和数据科学中最有力的工具之一。在负责推进工程和科学的大量领域的核心数据驱动技术中，它们是不可或缺的。虽然这些工具代表着统计学家发挥了根本性和决定性作用的显著成就，但前方仍有紧迫而艰巨的挑战。随着这些工具越来越多地应用于越来越大的数据集和系统，人们深感担忧的是，它们可能不再被理解，它们的构建和部署也不可靠或可靠。当仅仅被视为拟合密度和曲线的黑盒建模工具时，潜变量模型很难解释，当出现问题时，无论是模型严重错误指定还是学习算法简单故障，都很难检测或修复。该项目旨在解决现代潜变量模型中出现的理论和计算问题，以及这些统计模型在错误指定时的学习效率和可解释性。该项目的目标是为潜变量模型开发新的方法、算法和理论。它有三个主要目标：(1)潜在变量模型中出现的参数估计的统计理论；(2)可扩展的参数学习算法，它考虑了潜在结构的几何形状以及来自特定应用领域的数据表示的几何形状；(3)模型错误指定对参数估计的影响推动了新方法的发展。这三个被广泛描述的目标在一定程度上是由于PI在几个数据驱动的领域与科学家和工程师的合作努力，即智能交通、天体物理和信息提取的主题建模。在所有这些领域，潜变量模型作为一种有效的近似工具受到青睐，但实践者不仅对预测性能感兴趣，而且对可解释性感兴趣。在方法和工具上，本研究借鉴和贡献了统计学习、非参数贝叶斯统计和非凸优化等相关领域。在更广泛的影响方面，为潜变量模型开发可扩展的几何和变分推理算法，将有助于扩大在分析复杂和大数据时不可或缺的统计和计算工具箱。研究奇异结构的几何结构以及基于最优传输的理论在模型分析和算法开发中的作用，将有助于加速统计学与数学、计算机科学和运筹学之间的交叉交流。在教育和培训方面，该项目的跨学科性质提供了一个令人兴奋的机会，以吸引和培训一代研究人员和学生，学习变式方法和优化、统计学和数学，以及机器学习和智能基础设施。在这个项目中开发的材料将被整合到密歇根大学开发的本科生荣誉课程和统计科学和大数据分析暑期学校。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Latent variables models have become one of the most powerful tools in modern statistics and data science. They are indispensable in the core data-driven technologies responsible for advancing a vast array of domains of engineering and sciences. While these tools represent remarkable achievements in which statisticians have played fundamental and decisive roles, there are urgent and formidable challenges lying ahead. As these tools are increasingly applied to ever bigger data sets and systems, there are deep concerns that they may no longer be understood, nor is their construction and deployment reliable or robust. When treated as merely black-box modeling devices for fitting densities and curves, latent variable models are difficult to interpret and can be hard to detect or fix when something goes wrong, either when the model is severely misspecified or the learning algorithms simply break down. This project aims to address the theoretical and computational issues that arise in modern latent variable models, and the learning efficiency and interpretability of such statistical models when they are misspecified.The goals of this project are to develop new methods, algorithms and theory for latent variable models. There are three major aims: (1) a statistical theory for parameter estimation that arises in latent variable models; (2) scalable parameter learning algorithms which account for the geometry of the latent structures, as well as the geometry of the data representation arising from specific application domains; and (3) impacts of model misspecification on parameter estimation motivating the development of new methods. These three broadly described aims are partly motivated by the PI's collaborative efforts with scientists and engineers in several data-driven domains, namely intelligent transportation, astrophysics and topic modeling for information extraction. In all these domains, latent variable models are favored as an effective approximation device, but practitioners are interested in not only predictive performance but also interpretability. In terms of methods and tools, this research draws from and contributes to several related areas including statistical learning, nonparametric Bayesian statistics and non-convex optimization. In terms of broader impacts, the development of scalable geometric and variational inference algorithms for latent variable models will help to expand the statistical and computational tool box that are indispensable in the analysis of complex and big data. The investigation into the geometry of singularity structures and the role of optimal transport based theory in the analysis of models and the development of algorithms will help to accelerate the cross-fertilization between statistics and mathematics, computer science and operations research. In terms of education and training, the interdisciplinary nature of this project provides an exciting opportunity to attract and train a generation of researchers and students in variational methods and optimization, statistics and mathematics, as well as machine learning and intelligent infrastructure. The materials developed in this project will be integrated into an undergraduate honor course and a summer school for statistical science and big data analytics developed at the University of Michigan.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.