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Through a careful and rigorous treatment, we provide a better understanding of the operating characteristics of multiway PCA, develop improved statistical methods, and reveal new opportunities and challenges when moving from vector data to multiway data. 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 with a much smaller sample size requirement than the usual PCs. Furthermore, the sample multiway PCs are asymptotically independent and the asymptotic distribution of each sample PC does not depend any other PCs, 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.<\/jats:p>","DOI":"10.1214\/25-aos2508","type":"journal-article","created":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T05:22:56Z","timestamp":1757654576000},"page":"1431-1456","source":"Crossref","is-referenced-by-count":2,"title":["On the multiway principal component analysis"],"prefix":"10.1214","volume":"53","author":[{"given":"Jialin","family":"Ouyang","sequence":"first","affiliation":[{"name":"Department of Statistics, Columbia University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ming","family":"Yuan","sequence":"additional","affiliation":[{"name":"Department of Statistics, Columbia University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"22","unstructured":"<small>Hopkins, S. B.<\/small>, <small>Shi, J.<\/small> and <small>Steurer, D.<\/small> (2015). 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