Multiway data are becoming more and more common. While there are many approaches to extending principal component analysis (PCA) from usual data matrices to multiway arrays, their conceptual differences from the usual PCA, and the methodological implications of such differences remain largely unknown. This work aims to specifically address these questions. In particular, we clarify the subtle difference between PCA and singular value decomposition (SVD) for multiway data, and show that multiway principal components (PCs) can be estimated reliably in absence of the eigengaps required by the usual PCA, and in general much more efficiently than the usual PCs. Furthermore, the sample multiway PCs are asymptotically independent and hence allow for separate and more accurate inferences about the population PCs. The practical merits of multiway PCA are further demonstrated through numerical, both simulated and real data, examples.

翻译：多路数据正变得越来越常见。尽管已有多种方法将主成分分析（PCA）从常规数据矩阵扩展到多路数组，但这些方法与常规PCA的概念差异，以及这些差异对方法学的影响，在很大程度上仍不为人知。本研究旨在专门探讨这些问题。具体而言，我们阐明了多路数据中PCA与奇异值分解（SVD）之间的细微差别，并证明多路主成分（PCs）可以在常规PCA所需的特征间隙缺失的情况下被可靠估计，且通常远比常规PCs更为高效。此外，样本多路PCs渐近独立，因此能够对总体PCs进行分离且更准确的推断。通过数值模拟和真实数据示例，进一步展示了多路PCA的实践优势。