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On the asymptotic normality and efficiency of Kronecker envelope principal component analysis
Journal of Multivariate Analysis ( IF 1.6 ) Pub Date : 2021-04-18 , DOI: 10.1016/j.jmva.2021.104761
Shih-Hao Huang , Su-Yun Huang

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.



中文翻译:

关于Kronecker包络主成分分析的渐近正态性和有效性。

近年来,矩阵或张量数据的降维方法一直是活跃的研究领域。Li等。(2010年)介绍了Kronecker包络线的概念,并提出了用于监督尺寸缩减的尺寸折叠估算器。在低温电子显微镜(cryo-EM)图像的数据分析中(Chen等人,2014),使用Kronecker包络主成分分析(PCA)来减小低温EM图像的尺寸。Kronecker包络PCA是一个分为两步的过程,第一步是将数据投影到多线性包络子空间上,然后在投影的核心张量上使用普通PCA。当搜索领先的主要子空间时,多线性包络子空间保留了观测值的自然Kronecker乘积结构。保留Kronecker乘积结构的主要优点是在指定领先的主子空间时可以简化参数的使用,从而减轻了高维的不利影响。PCA方法将可能相关的变量转换为不相关的变量,并进一步减小投影核心张量的尺寸。在本文中,我们得出了Kronecker包络PCA的渐近正态性,并将其与普通PCA进行了比较。利用专业化理论,我们证明了Kronecker包络PCA的渐近效率比普通PCA更高,因为Kronecker包络PCA的渐近总随机变化小于普通PCA。提出了一个富有启发性的低温EM图像聚类和仿真研究的实际数据示例,以显示Kronecker信封PCA的优点。

更新日期:2021-04-26
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