Nonlinear Factor and Latent Variable Models

本研究项目将开创新的非线性降维数据分析方法。这些方法将被设计成从大量观察到但可能有错误污染的数据中提取关于感兴趣的几个特定的未观察到的变量的信息。这种类型的非线性方法本身就具有挑战性，因为存在这样的风险，即数据本身可能不能唯一地识别感兴趣变量的特征，除非作出谨慎的激励假设并执行详细的形式分析。新的非线性方法将推广目前在统计调查、经济、医学成像、数据压缩、机器学习和互联网搜索引擎中广泛使用的线性降维统计分析方法。开发这些方法的有效非线性扩展将大大提高这些领域的数据分析能力，使研究人员能够揭示目前被线性方法的透镜掩盖的错综复杂的非线性关系。研究生将通过在新方法的开发和实施中发挥重要作用而获得有价值的研究、教育和培训。由该项目开发的软件将公之于众。在经济学和统计学中，大量甚至是不完全测量的变量所包含的信息可以用少数变量(“因素”或“主成分”)来概括的想法已被广泛采用，并随着“大数据”的日益普及而受到越来越多的关注。一些待开发的方法可以被视为广泛使用的技术类别的统一和推广，例如线性潜在因子模型、多向阵列分解和非经典测量误差模型。其他待开发的方法将广泛使用的线性主成分分析推广到非线性设置。这项工作涉及的主要问题是：什么合理的假设确保许多观测变量的分布唯一地确定感兴趣的特定未观测变量的分布？能否设计出有效的数值算法来找到观测到的分布和未观测到的分布之间的这种唯一映射？在特殊情况下，新方法是否会简化为现有方法？用来回答这些问题的方法来自于最优运输、熵最大化、算子理论和测度论等领域。

This research project will create new nonlinear dimension-reducing data analysis methods. The methods will be designed to extract information regarding a few specific unobserved variables of interest from large amounts of observed but potentially error-contaminated data. This type of nonlinear approach is inherently challenging due to the risk that the data alone may not uniquely identify the features of the variables of interest unless carefully motivated assumptions are made and a detailed formal analysis is performed. The new nonlinear methods will generalize current linear dimension-reducing statistical analysis methods that are widely used in statistical surveys, economics, medical imaging, data compression, machine learning, and internet search engines. Developing efficient nonlinear extensions of these methods will advance data analysis capabilities in these fields considerably by enabling researchers to uncover intricate nonlinear relationships that are currently masked through the lenses of linear approaches. A graduate student will obtain valuable research education and training by playing an important role in the development and implementation of the new methods. Software developed from this project will be made publicly available.The idea that the information contained in a large number of even imperfectly measured variables can be summarized by a small number of variables (the "factors" or "principal components") has been widely adopted in economics and statistics and is gathering even more attention with the increasing prevalence of "big data." Some of the methods to be developed can be seen as a unification and generalization of widely used classes of techniques, such as linear latent factor models, multiway array decomposition, and nonclassical measurement error models. Other methods to be developed generalize the widely used linear principal component analysis to nonlinear settings. The main questions addressed in this work are: What reasonable assumptions ensure that the distribution of the many observed variables uniquely determine the distribution of the specific unobserved variables of interest? Can efficient numerical algorithms be devised to find this unique mapping between observed and unobserved distributions? Do the new methods reduce to existing methods in special cases? The approaches used to answer these questions draw from the fields of optimal transport, entropy maximization, operator theory, and measure theory.