Facing with rapidly increasing demands for analyzing high-order data or multiway data, feature-extracting methods become imperative for analysis and processing. The traditional feature-extracting methods, however, either need to overly vectorize the data and smash the original structure hidden in data, such as PCA and PCA-like methods, which is unfavorable to the data recovery, or cannot eliminate the redundant information very well, such as tucker decomposition (TD) and TD-like methods. To overcome these limitations, we propose a more flexible and more powerful tool, called the multiview principal components analysis (Multiview-PCA) in this article. By segmenting a random tensor into equal-sized subarrays called sections and maximizing variations caused by orthogonal projections of these sections, the Multiview-PCA finds principal components in a parsimonious and flexible way. In so doing, two new operations on tensors, the S -direction inner/outer product, are introduced to formulate tensor projection and recovery. With different segmentation ways characterized by section depth and direction, the Multiview-PCA can be implemented many times in different ways, which defines the sequential and global Multiview-PCA, respectively. These multiple Multiview-PCA take the PCA and PCA-like, and TD and TD-like as the special cases, which correspond to the deepest section depth and the shallowest section depth, respectively. We propose an adaptive depth and direction selection algorithm for the implementation of Multiview-PCA. The Multiview-PCA is then tested in terms of subspace recovery ability, compression ability, and feature extraction performance when applied to a set of artificial data, surveillance videos, and hyperspectral imaging data. All numerical results support the flexibility, effectiveness, and usefulness of Multiview-PCA.

Feature screening is a powerful tool in processing high-dimensional data. When the sample size N and the number of features p are both large, the implementation of classic screening methods can be numerically challenging. In this paper, we propose a distributed screening framework for big data setup. In the spirit of 'divide-and-conquer', the proposed framework expresses a correlation measure as a function of several component parameters, each of which can be distributively estimated using a natural U-statistic from data segments. With the component estimates aggregated, we obtain a final correlation estimate that can be readily used for screening features. This framework enables distributed storage and parallel computing and thus is computationally attractive. Due to the unbiased distributive estimation of the component parameters, the final aggregated estimate achieves a high accuracy that is insensitive to the number of data segments m. Under mild conditions, we show that the aggregated correlation estimator is as efficient as the centralized estimator in terms of the probability convergence bound and the mean squared error rate; the corresponding screening procedure enjoys sure screening property for a wide range of correlation measures. The promising performances of the new method are supported by extensive numerical examples.