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.

中文翻译：

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Multiview PCA：一种高阶数据的特征提取和降维方法

面对快速增长的分析高阶数据或多路数据的需求，特征提取方法对于分析和处理变得势在必行。而传统的特征提取方法要么需要对数据进行过度向量化，破坏隐藏在数据中的原始结构，如PCA和类PCA方法，不利于数据恢复，要么不能很好地去除冗余信息。 ，例如塔克分解（TD）和类似TD的方法。为了克服这些限制，我们提出了一种更灵活、更强大的工具，在本文中称为多视图主成分分析 (Multiview-PCA)。通过将随机张量分割成大小相等的子数组，称为截面和最大化这些正交投影引起的变化部分，Multiview-PCA 以简洁和灵活的方式找到主要组件。这样做，张量上的两个新操作， $新元 - 方向内/外积，用于制定张量投影和恢复。具有不同的分割方式，其特点是截面深度和方向，Multiview-PCA 可以以不同的方式实现多次，分别定义了顺序和全局 Multiview-PCA。这些多重Multiview-PCA以PCA和PCA-like、TD和TD-like为特例，分别对应最深截面深度和最浅截面深度。我们提出了一种自适应深度和方向选择算法，用于实现 Multiview-PCA。然后，当应用于一组人工数据、监控视频和高光谱成像数据时，Multiview-PCA 在子空间恢复能力、压缩能力和特征提取性能方面进行了测试。所有数值结果都支持 Multiview-PCA 的灵活性、有效性和实用性。