Collaborative Research: Statistical Methods, Algorithms, and Theory for Large Tensors

合作研究：大张量的统计方法、算法和理论

• 批准号: 1803450
• 负责人: Ming Yuan
• 金额: $ 28万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing 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=1803450&HistoricalAwards=false
• 关键词: Collaborative; Research; Statistical; Methods; Algorithms

Large amounts of multidimensional data in the form of multilinear arrays, or tensors, arise routinely in modern applications from such diverse fields as chemometrics, genomics, physics, psychology, and signal processing, among many others. At the present time, our ability to generate and acquire such data has far outpaced our ability to effectively extract useful information. There is a clear need to develop novel statistical methods, efficient computational algorithms, and fundamental mathematical theory to analyze and exploit information in these types of data. This research project builds upon prior work in high-dimensional statistics, genomics, quantum physics, fast functional MRI, and closed-loop diabetes control to address the challenges in analysis of large tensorial data sets. It is anticipated that the project will help to advance future research in these and other areas of applications.One of the main challenges in dealing with this type of data is to develop methods of statistical inference to achieve both the statistical and computational efficiencies. More often than not, these two aims are not simultaneously achieved by existing methods: there is a gap between the optimal rate in the minimax sense and the best rate achievable by existing polynomial-time algorithms. The overarching goal of this project is to develop statistical methods, algorithms, and theory to efficiently, both statistically and computationally, analyze large scale data in the form of tensors. In particular, four most common and interrelated problems will be studied systematically: low-rank tensor denoising, low-rank tensor regression, estimation of the moment tensor, and tensor phase retrieval.

大量的多维数据以多线性阵列或张量的形式出现在现代应用中，这些应用来自化学计量学，基因组学，物理学，心理学和信号处理等许多领域。目前，我们生成和获取此类数据的能力远远超过了我们有效提取有用信息的能力。显然需要开发新的统计方法，有效的计算算法和基本的数学理论来分析和利用这些类型的数据中的信息。该研究项目建立在高维统计，基因组学，量子物理，快速功能性MRI和闭环糖尿病控制方面的先前工作的基础上，以解决大型张量数据集分析中的挑战。 预计该项目将有助于推进这些和其他应用领域的未来研究。处理这类数据的主要挑战之一是开发统计推断方法，以实现统计和计算效率。通常情况下，这两个目标是不能同时实现现有的方法：在极大极小意义上的最佳速率和现有的多项式时间算法可实现的最佳速率之间存在差距。该项目的总体目标是开发统计方法，算法和理论，以有效地在统计和计算上分析张量形式的大规模数据。特别是，四个最常见的和相互关联的问题将系统地研究：低秩张量去噪，低秩张量回归，估计的矩张量，和张量相位恢复。

项目成果

Non-Convex Projected Gradient Descent for Generalized Low-Rank Tensor Regression

• 发表时间: 2016-11
• 期刊: J. Mach. Learn. Res.
• 作者: Han Chen;Garvesh Raskutti;M. Yuan

Effective Tensor Sketching via Sparsification

• DOI: 10.1109/tit.2021.3049174
• 发表时间: 2017-10
• 期刊: IEEE Transactions on Information Theory
• 影响因子: 2.5
• 作者: Dong Xia;M. Yuan

ISLET: Fast and Optimal Low-rank Tensor Regression via Importance Sketching

• DOI: 10.1137/19m126476x
• 发表时间: 2019-11
• 期刊: ArXiv
• 作者: Anru R. Zhang;Yuetian Luo;Garvesh Raskutti;M. Yuan

Global testing against sparse alternatives under Ising models

• DOI: 10.1214/17-aos1612
• 发表时间: 2016-11
• 期刊: The Annals of Statistics
• 作者: R. Mukherjee;S. Mukherjee;Ming Yuan

Statistical inferences of linear forms for noisy matrix completion

噪声矩阵完成的线性形式的统计推断

• DOI: 10.1111/rssb.12400
• 发表时间: 2021
• 期刊: Journal of the Royal Statistical Society: Series B (Statistical Methodology
• 作者: Xia, Dong;Yuan, Ming
• 通讯作者: Yuan, Ming