CAREER: Computer-Intensive Statistical Inference on High-Dimensional and Massive Data: From Theoretical Foundations to Practical Computations

在大数据时代，计算机密集型统计推理面临着前所未有的挑战和机遇。高维和海量数据现在出现在科学领域，包括生物医学工程、环境科学、金融计量经济学、阵列信号处理和社会网络等。一个重要的相关研究挑战是开发有效的方法来提取信息并量化其对大量变量和测量的不确定性。对于高维大规模数据集，通过Bootstrap方法进行不确定性量化的数据驱动的统计推理过程通常是计算密集型的。在计算方面，本研究项目将通过并行高性能计算技术利用分布式推理，这是加速Bootstrap计算的重要组成部分。在统计方面，这项研究将介绍一个研究各种Bootstrap方法的性能的一般框架。这一研究项目的目的是全面了解统计和计算之间的基本权衡，以量化一大类推理程序的不确定性，从而为在潜在的实际应用中实际优化统计精度和计算成本提供指导。这项研究的主要目标是提供新的见解，并加深对高维和海量数据框架中完全依赖数据的推理过程(如Bootstraps)在两个经典问题上的优势和基本局限性的理论理解：i)变点检测和识别；ii)U-统计的计算感知统计推断。这项研究的目的是在维度可以比样本大小更大(甚至更大)的情况下，开发统计上正确的、计算上可扩展的推理程序。与现有的工作相比，正在开发的方法具有强大的理论保证，在温和的假设下是健壮的，不需要调整，并且很容易并行化。具有实际意义的是，这项研究将为应用高维和非参数统计的学科的研究人员开发所需的软件工具。所提出的研究的理论贡献包括在高维和无限维空间中建立了新的逼近和耦合定理(在比现有文献更弱的正则性条件下)，这些空间的维度和复杂性不断增加，其中经典的概率工具如中心极限定理和极值理论不再适用。数学理论具有独立的意义，并将提供强大的新工具来分析高维和非参数模型上的其他统计程序。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。

In an era of Big Data, computer-intensive statistical inference faces unprecedented challenges and opportunities. High-dimensional and massive data are now emerging in scientific areas including biomedical engineering, environmental science, financial econometrics, array signal processing, and social networks, among many others. An important associated research challenge is to develop efficient methods to extract information and quantify its uncertainty for a large number of variables and measurements. Data-driven statistical inferential procedures for uncertainty quantification via the bootstrap methods are often computationally intensive for high-dimensional large-scale datasets. On the computational side, this research project will make use of distributed inference via the parallel high-performance computing technique, which is an essential ingredient to speed up bootstrap calculations. On the statistical side, this research will introduce a general framework for studying the performance of various bootstrap methods. This research project aims to lead to a comprehensive understanding of the fundamental tradeoff between statistical and computational concerns in quantifying uncertainty for a broad class of inferential procedures, thus providing guidance to practically optimize statistical accuracy and computational cost in potential real applications. Both undergraduate and graduate students are involved in the project.The overarching goal of this research project is to provide new insights and deepen the theoretical understanding of strengths and fundamental limitations of fully data-dependent inferential procedures (such as bootstraps) in the high-dimensional and massive data framework on two classical problems: i) change point detection and identification; ii) computationally-aware statistical inference for U-statistics. The research aims to develop statistically correct and computationally scalable inferential procedures when the dimension can be larger (or even much larger) than the sample size. In contrast to existing work, the methods under development have strong theoretical guarantees, are robust under mild assumptions, require no tuning, and are easy to parallelize. Of practical interest, the research will develop needed software tools for researchers from disciplines with applications of high-dimensional and nonparametric statistics. Theoretical contributions of the proposed research include establishing new approximation and coupling theorems (under weaker regularity conditions than existing literature) in high-dimensional and infinite-dimensional spaces of increasing dimension and complexity, where classical probability tools such as the central limit theorem and extreme value theory are no longer applicable. The mathematical theory is of independent interest and will provide powerful new tools to analyze other statistical procedures on high-dimensional and nonparametric models.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.