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

在化学计量学、基因组学、物理学、心理学和信号处理等不同领域的现代应用中，经常会出现大量多线性数组或张量形式的多维数据。目前，我们生成和获取此类数据的能力已经远远超过了我们有效提取有用信息的能力。显然需要开发新颖的统计方法、高效的计算算法和基础数学理论来分析和利用这些类型数据中的信息。该研究项目建立在高维统计、基因组学、量子物理学、快速功能 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。处理此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，现有方法无法同时实现这两个目标：极小极大意义上的最优速率与现有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，以便在统计和计算上有效地分析张量形式的大规模数据。特别是，将系统地研究四个最常见且相互关联的问题：低秩张量去噪、低秩张量回归、矩张量估计和张量相位检索。

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.