Dimension reduction for high-dimensional high-order data

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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.