Statistical Machine Learning Methods for Complex Data Sets

科学技术的最新进展导致了大量复杂结构的大规模数据的产生，包括基因组学、神经影像学和微生物学数据。这些大规模数据集对数据分析提出了重大的统计和计算挑战。首先，广泛使用的统计方法会产生不稳定的估计，并且无法在计算上扩展到大规模数据集的建模。其次，由于测量误差或重尾随机噪声的存在，复杂的数据集往往伴随着异常值。例如，在基因组研究中，我们观察到基因表达水平的分布通常是重尾的，即数据中包含了很多极大的值。如果在模型估计和推理过程中不考虑这些异常值，经典的统计方法将产生有偏差的估计和虚假的科学发现。该项目旨在开发可扩展和健壮的多元统计方法来解决上述问题。在这个项目中，研究者使用正则化和统计优化技术的组合来开发新的多元统计方法来分析复杂的高维数据集。项目的第一部分涉及稀疏广义特征值问题，该问题自然出现在许多统计模型中，如偏最小二乘、典型相关分析、充分降维和费雪判别分析。研究者将开发一个解决稀疏广义特征值问题的通用框架，并为分析高维数据提供广泛的统计模型。此外，研究者将研究稀疏广义特征值问题的理论性质，这将导致对各种统计模型的理解，这些模型以前在高维环境中没有得到很好的理解。研究项目的第二部分着重于一类鲁棒稀疏降阶回归模型。研究者将为Huber损失函数下的结果估计器开发有效的算法和高维渐近分析，并量化使用Huber损失和平方误差损失之间的偏差-鲁棒权衡。该研究项目还将提供易于使用的软件包，以适应开发的方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Recent advances in science and technology have led to the generation of massive amounts of large-scale data with complex structures, including genomics, neuroimaging, and microbiology data. These large-scale datasets pose significant statistical and computational challenges to data analysis. Firstly, widely used statistical methods yield unstable estimates and are not computationally scalable to modeling large-scale data sets. Secondly, complex data sets are often accompanied by outliers due to possibly measurement error or heavy-tailed random noise. For instance, in genomic studies, it has been observed that the distribution of gene expression levels is generally heavy-tailed, that is, the data contain a lot of extremely large values. Classical statistical methods will yield biased estimates and spurious scientific discovery if these outliers are not taken into account during model estimation and inference. This project aims to develop scalable and robust multivariate statistical methods to address the aforementioned problems. In this project, the investigator uses a combination of regularization and statistical optimization techniques to develop novel multivariate statistical methods for analyzing complex high-dimensional data sets. The first part of the project concerns the sparse generalized eigenvalue problem, which arises naturally in many statistical models such as partial least squares, canonical correlation analysis, sufficient dimension reduction, and Fisher's discriminant analysis. The investigator will develop a general framework for solving the sparse generalized eigenvalue problem and make available a wide range of statistical models for analyzing high-dimensional data. Furthermore, the investigator will study the theoretical properties of sparse generalized eigenvalue problem, and this will lead to the understanding of various statistical models that are previously not well understood in the high-dimensional setting. The second part of the research project focuses on a class of robust sparse reduced rank regression models. The investigator will develop efficient algorithms and high-dimensional asymptotic analysis for the resulting estimators under the Huber loss function, and quantify the bias-robust tradeoff between using Huber loss and squared error loss. This research project will also deliver easy-to-use software packages for fitting the developed methods.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.