Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Model

In modern scientific applications, data are often observed in the form of multiple matrices or tensors that pertain to different subjects from a certain population. For instance, longitudinal gene expression data consist of a matrix of gene expression levels across time for each subject (Liu et al., 2017); MRI imaging data contain one order-3 tensor image for each patient (Zhou et al., 2013); multilayer network can be represented by an order-3 tensor, where each layer (i.e., a matrix) represents one network (Jing et al., 2021); m-uniform hypergraph is typically viewed as an order-m tensor, whose entries denote all hyper-edges (Zhen & Wang, 2022); atomicresolution 4D scanning transmission electron microscopy data can be expressed as an order-3 tensor with two models denoting scan location and the other denoting the convergent beam electron diffraction pattern (Zhang et al., 2020). Combining information from all subjects results in a high-order tensor with subject independence along one mode and some covariance structure along the other modes that represent the relationship among the measured covariates. Principal Component Analysis (PCA) is a widely accepted method for analyzing data consisting of vectors associated with individual subjects. Its primary objective is to identify a lower-dimensional subspace within the feature domain that captures the majority of data variance (Pearson, 1901).