Dimension reduction for high-dimensional high-order data

高维、高阶数据通常来自于广泛的应用，例如神经成像、微生物学、生物信息学、纵向研究和材料科学。这些数据与传统的低维或低阶数据相比具有鲜明的特点，对统计界提出了前所未有的挑战。为了更好地总结、可视化和分析高维高阶数据，降维往往成为关键的第一步。一种自然的方法是将高阶数据展开为矩阵，然后使用成熟的矩阵降维方法。然而，这样的操作经常导致关于内在数据结构的信息的丢失和后续分析的次优统计结果。此外，传统统计方法对高维、高阶数据的简单推广通常在统计或计算上是不可行的。因此，迫切需要新的高维高阶数据降维方法，在本项目中，PI计划开发新的理论，方法和计算算法，以解决高维高阶数据降维中的一系列基础问题。该项目将侧重于四个主要领域：（一）正则化张量奇异值分解的一般框架;（二）通过交替退火可证明的正则化张量分解;（三）由新的高阶尖峰协方差模型提供理论保证的高阶PCA;（四）通过张量方法进行动态网络社区检测。该奖项反映了NSF的法定使命，并通过使用基金会的学术价值和更广泛的影响评审标准。

High-dimensional, high-order data commonly arise from a wide range of applications, such as neuroimaging, microbiology, bioinformatics, longitudinal studies, and material sciences. These data possess distinct characteristics compared with traditional low-dimensional or low-order data and pose unprecedented challenges to the statistical community. Dimension reduction often becomes the crucial first step in order to better summarize, visualize, and analyze high-dimensional high-order data. One natural approach is to unfold high-order data into matrices, followed by the use of well-established matrix dimension reduction methods. However, such operations often lead to loss of information on the intrinsic data structures and sub-optimal statistical results of subsequent analyses. In addition, the naive generalization of traditional statistical methods to high-dimensional, high-order data is often statistically or computationally infeasible. Therefore, there is a critical need for new high-dimensional high-order data dimension reduction methods.In this project, the PI plans to develop new theories, methodologies, and computational algorithms to address a series of fundamental problems in high-dimensional, high-order data dimension reduction. The project will be focused on four major areas: (i) a general framework for regularized tensor SVD; (ii) provable regularized tensor decomposition via alternating annealing; (iii) high-order PCA with theoretical guarantees by a new high-order spiked covariance model; (iv) dynamic network community detection via tensor methods.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.