On the asymptotic normality and efficiency of Kronecker envelope principal component analysis

Dimension reduction methods for matrix or tensor data have been an active research field in recent years. Li et al. (2010) introduced the notion of the Kronecker envelope and proposed dimension folding estimators for supervised dimension reduction. In a data analysis of cryogenic electron microscopy (cryo-EM) images (Chen et al., 2014), Kronecker envelope principal component analysis (PCA) was used to reduce the dimension of cryo-EM images. Kronecker envelope PCA is a two-step procedure, which consists of projecting data onto a multilinear envelope subspace as the first step, followed by ordinary PCA on the projected core tensor. The multilinear envelope subspace preserves the natural Kronecker product structure of observations when searching for the leading principal subspace. The main advantage of preserving the Kronecker product structure is the parsimonious usage of parameters in specifying the leading principal subspace, which mitigates the adverse influence of high-dimensionality. The method of PCA will convert possibly correlated variables to uncorrelated ones and further reduce the dimension of the projected core tensor. In this article we derive the asymptotic normality of Kronecker envelope PCA and compare it with ordinary PCA. Utilizing majorization theory, we show that Kronecker envelope PCA is asymptotically more efficient than ordinary PCA in the sense that the asymptotic total stochastic variation of Kronecker envelope PCA is smaller than that of ordinary PCA. A motivating real data example of cryo-EM image clustering and simulation studies are presented to show the merits of Kronecker envelope PCA.