CAREER: Something Old, Something New: Robust Statistics in the 21st Century

当代涉及大型高维数据集的科学问题为统计学家带来了一个新时代。一个在很大程度上未被探索的研究领域涉及将稳健统计的想法纳入越来越多的高维估计和推断方法集合中。初步结果很有希望，但当试图将概念从经典稳健统计推广到高维环境时，大量的理论和哲学挑战比比皆是。研究人员将为高维稳健估计开发新的统计方法，并为所建议的估计器推导严格的理论。这个研究项目本质上是高度跨学科的，横跨统计学、工程学和计算机科学。研究结果将被广泛传播，导致不同领域之间的交叉授粉，并重新激发人们对稳健统计的兴趣。此外，研究人员将改进和测试她在放射学应用中的方法，推动新的科学合作，并导致更强大的医学成像程序部署在医学研究中。研究人员还将在这项研究的基础上开发新的教育材料，这些材料将被纳入研究生和本科生水平的机器学习交叉列表课程。研究人员将通过公开演讲和访问威斯康星州的高中数学圈，努力改善统计和数据科学的形象。这项研究项目中要探索的问题包括：(1)现有的稳健性概念如何应用于高维环境？(2)应该如何修改高维估计程序，以防止偏离分布假设？(3)如何量化各种提案的相对稳健性？该项目旨在产生新的理论成果，推动统计学和最优化理论的前沿。将为高维统计估计设计新的算法，在更广泛的一组模型假设下保证精度。理论分析将涉及研究优化社区中各种独立感兴趣的非凸估计，重点是产生统计上一致解的目标和优化算法。该研究项目还将解决稳健统计学中长期悬而未决的问题，涉及低维非凸目标函数的优化，研究人员将检查机器学习应用中出现的各种新问题设置，包括恶意污染数据、非IID观察和被错误标记为训练和测试数据的数据集。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。

Contemporary scientific problems involving large, high-dimensional datasets have ushered in a new era for statisticians. A largely unexplored area of research concerns incorporating ideas from robust statistics into the growing collection of methods for high-dimensional estimation and inference. Preliminary results hold much promise, but a plethora of theoretical and philosophical challenges abound when attempting to generalize notions from classical robust statistics to high-dimensional settings. The investigator will develop new statistical methodology for high-dimensional robust estimation and derive rigorous theory for the proposed estimators. This research project is highly interdisciplinary in nature, cutting across statistics, engineering, and computer science. Results of the research will be disseminated broadly, leading to cross-pollination between fields and revitalized interest in robust statistics. In addition, the investigator will refine and test her methods in radiology applications, instigating new scientific collaborations and leading to more robust medical imaging procedures for deployment in medical research. The investigator will also develop new educational material based on the research that will be incorporated into cross-listed classes in machine learning at the graduate and undergraduate levels. The investigator will work to improve the image of statistics and data science by engaging the wider community through public speaking engagements and visits to high school math circles across the state of Wisconsin.Questions to be explored in this research project include: (1) How do existing notions of robustness apply to high-dimensional settings? (2) How should high-dimensional estimation procedures be modified to protect against deviations from distributional assumptions? (3) How might one quantify the relative robustness of various proposals? The project aims to generate novel theoretical results that advance the frontiers of both statistics and optimization theory. New algorithms will be devised for high-dimensional statistical estimation with guaranteed accuracy under a broader set of model assumptions. The theoretical analysis will involve studying a variety of non-convex estimators of independent interest in the optimization community, with emphasis on objectives and optimization algorithms that give rise to statistically consistent solutions. The research project will also address long-standing open questions in robust statistics involving optimization of low-dimensional non-convex objective functions, and the investigator will examine a variety of new problem settings arising in machine learning applications, including adversarially contaminated data, non-iid observations, and mislabeled datasets dichotomized into training and testing data.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.