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

涉及大型高维数据集的当代科学问题为统计学家开创了一个新时代。一个很大程度上尚未探索的研究领域涉及将稳健统计的思想纳入不断增长的高维估计和推理方法中。初步的结果有很大的希望，但大量的理论和哲学的挑战比比皆是时，试图概括的概念，从经典的强大的统计高维设置。研究人员将开发新的统计方法的高维鲁棒估计，并推导出严格的理论提出的估计。这个研究项目是高度跨学科的性质，跨越统计，工程和计算机科学。研究结果将广泛传播，使各领域之间相互借鉴，并重新激发人们对健全的统计数据的兴趣。此外，研究人员将完善和测试她在放射学应用中的方法，促进新的科学合作，并导致更强大的医学成像程序在医学研究中的部署。研究人员还将根据研究开发新的教育材料，这些材料将被纳入研究生和本科生机器学习的交叉课程中。研究者将通过公开演讲活动和访问威斯康星州的高中数学圈来提高统计和数据科学的形象。在这个研究项目中要探索的问题包括：（1）现有的鲁棒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)如何量化各种建议的相对稳健性？该项目旨在产生新的理论成果，推进统计学和优化理论的前沿。新的算法将被设计用于高维统计估计，并在更广泛的模型假设下保证准确性。理论分析将涉及研究各种非凸估计的独立利益的优化社区，重点是目标和优化算法，产生统计上一致的解决方案。该研究项目还将解决鲁棒统计中长期存在的公开问题，涉及低维非凸目标函数的优化，研究人员将研究机器学习应用中出现的各种新问题，包括受不利污染的数据，非iid观测，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响审查标准。

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.