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Bridging the Gap Between Theory and Applications: Robust and Scalable Statistical Estimation

弥合理论与应用之间的差距：稳健且可扩展的统计估计

基本信息

批准号：
1712956
负责人：
Stanislav Minsker
金额：
$ 10万
依托单位：
University of Southern California
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712956&HistoricalAwards=false
关键词：
Bridging Gap Between Theory Applications

项目摘要

Major challenges in modern data-rich environment require new statistical algorithms that succeed under realistic scenarios and model assumptions, such as estimation in the distributed setting, ability to handle heavy-tailed data, outliers, and missing observations. Research that will be performed by the Principal Investigator (PI) in the course of this project focuses on two important problems faced by contemporary statistical science: scalability and robustness. The goal of the project is to advance our understanding of statistical techniques that involve (a) high-dimension covariance matrix estimation, and (b) distributed statistical estimation protocols. Obtained results will be of interest to scientists working on theory as well as applications.One part of this project aims at answering open questions related to high-dimensional covariance matrix estimation for the heavy-tailed distributions. Such distributions serve as a viable model for data corrupted with outliers, an almost inevitable scenario in applications. Covariance matrix is one of the most fundamental objects in high-dimensional data analysis: many important statistical tools, such as Principal Component Analysis (PCA) and regression analysis, involve covariance estimation as a crucial step. For instance, PCA has striking connections to nonlinear dimension reduction and manifold learning techniques, genetics, computational biology, among many others. However, the assumptions underlying the theoretical analysis of most existing estimators, such as various modifications of the sample covariance matrix, are often restrictive and do not hold for real-world scenarios. Using tools from the random matrix theory, the PI will develop a new class of robust estimators that are numerically tractable, show good practical performance and enjoy strong theoretical guarantees under much weaker conditions than currently available alternatives. Specifically, the goal of the project is to design estimators that admit tight concentration around the unknown "true" covariance matrix under weak assumptions on the underlying distribution, such as existence of moments of only low order. Another part of this project is devoted to novel algorithms for scalable estimation that can take advantage of the "divide and conquer" approach. Divide and conquer paradigm assumes that data is stored and analyzed in a distributed way by a cluster consisting of several machines: each of the machines in a cluster works on its own sub-sample while communication among different machines is limited, and final results are obtained by piecing the outcomes of these distributed computations together. The PI will develop a class of new divide and conquer strategies supported by strong theoretical evidence. The project will investigate connections between the distributed estimation strategies and robustness of resulting algorithms -- an important characteristic of large distributed systems.

现代数据丰富环境中的主要挑战需要新的统计算法，这些算法在现实场景和模型假设下取得成功，例如分布式环境中的估计，处理重尾数据，离群值和缺失观测的能力。主要研究者（PI）将在本项目过程中进行的研究集中在当代统计科学面临的两个重要问题：可扩展性和鲁棒性。该项目的目标是提高我们对统计技术的理解，包括（a）高维协方差矩阵估计，和（B）分布式统计估计协议。所得结果将对从事理论研究和应用研究的科学家们产生兴趣。本项目的一部分旨在回答与重尾分布的高维协方差矩阵估计相关的开放性问题。这样的分布作为一个可行的模型，数据损坏与离群值，在应用程序中几乎不可避免的情况。协方差矩阵是高维数据分析中最基本的对象之一，许多重要的统计工具，如主成分分析（PCA）和回归分析，都涉及协方差估计作为一个关键步骤。例如，PCA与非线性降维和流形学习技术、遗传学、计算生物学等有着惊人的联系。然而，大多数现有估计量的理论分析所依据的假设，如样本协方差矩阵的各种修改，往往是限制性的，并不适用于现实世界的情况。使用随机矩阵理论的工具，PI将开发一类新的鲁棒估计量，这些估计量在数值上易于处理，具有良好的实际性能，并且在比目前可用的替代方案弱得多的条件下具有强有力的理论保证。具体来说，该项目的目标是设计估计，承认周围的未知的“真正的”协方差矩阵的弱假设下的基础分布，如存在的时刻，只有低阶的紧浓度。这个项目的另一部分是致力于新的算法，可扩展的估计，可以利用“分而治之”的方法。分而治之范式假设数据由多台机器组成的集群以分布式方式存储和分析：集群中的每台机器都在自己的子样本上工作，而不同机器之间的通信是有限的，最终结果是通过将这些分布式计算的结果拼凑在一起来获得的。PI将开发一类新的分而治之的战略，由强有力的理论证据支持。该项目将研究分布式估计策略和由此产生的算法的鲁棒性之间的联系-大型分布式系统的一个重要特征。