{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,27]],"date-time":"2026-10-27T08:18:18Z","timestamp":1793089098091,"version":"4.5.0"},"reference-count":37,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T00:00:00Z","timestamp":1700438400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/100007015","name":"University of Wisconsin\u2013Madison","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100007015","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Office of the Vice Chancellor for Research and Graduate Education"},{"DOI":"10.13039\/100001395","name":"Wisconsin Alumni Research Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100001395","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,5,13]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Varimax factor rotations, while popular among practitioners in psychology and statistics since being introduced by Kaiser (1958), have historically been viewed with skepticism and suspicion by some theoreticians and mathematical statisticians. Now, work by Rohe &amp; Zeng (2023) provides new, fundamental insight: varimax rotations provably perform statistical estimation in certain classes of latent variable models when paired with spectral-based matrix truncations for dimensionality reduction. We build on this new-found understanding of varimax rotations by developing further connections to network analysis and spectral methods rooted in entrywise matrix perturbation analysis. Concretely, this paper establishes the asymptotic multivariate normality of vectors in varimax-transformed Euclidean point clouds that represent low-dimensional node embeddings in certain latent space random graph models. We address related concepts including network sparsity, data denoising and the role of matrix rank in latent variable parameterizations. Collectively, these findings, at the confluence of classical and contemporary multivariate analysis, reinforce methodology and inference procedures grounded in matrix factorization-based techniques. Numerical examples illustrate our findings and supplement our discussion.<\/jats:p>","DOI":"10.1093\/biomet\/asad061","type":"journal-article","created":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T20:33:49Z","timestamp":1700512429000},"page":"609-623","source":"Crossref","is-referenced-by-count":4,"title":["On varimax asymptotics in network models and spectral methods for dimensionality reduction"],"prefix":"10.1093","volume":"111","author":[{"given":"J","family":"Cape","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Wisconsin\u2013Madison , 1300 University Avenue , Madison, Wisconsin 53706, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,11,20]]},"reference":[{"key":"2024051307572568800_asad061-B1","doi-asserted-by":"crossref","first-page":"1452","DOI":"10.1214\/19-AOS1854","article-title":"Entrywise eigenvector analysis of random matrices with low expected rank","volume":"48","author":"Abbe","year":"2020","journal-title":"Ann. Statist"},{"key":"2024051307572568800_asad061-B2","doi-asserted-by":"crossref","first-page":"10101","DOI":"10.1073\/pnas.97.18.10101","article-title":"Singular value decomposition for genome-wide expression data processing and modeling","volume":"97","author":"Alter","year":"2000","journal-title":"Proc. Nat. Acad. Sci"},{"key":"2024051307572568800_asad061-B3","volume-title":"An Introduction to Multivariate Statistical Analysis","author":"Anderson","year":"1962","edition":"3rd ed"},{"key":"2024051307572568800_asad061-B4","first-page":"111","volume-title":"Proc. 3rd Berkeley Symp. Math. Statist. Prob","author":"Anderson","year":"1956","edition":"5, Ed"},{"key":"2024051307572568800_asad061-B5","doi-asserted-by":"crossref","first-page":"e8124","DOI":"10.15252\/msb.20178124","article-title":"Multi-omics factor analysis: a framework for unsupervised integration of multi-omics data sets","volume":"14","author":"Argelaguet","year":"2018","journal-title":"Molec. Syst. Biol"},{"key":"2024051307572568800_asad061-B6","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/s13171-021-00268-x","article-title":"Eigenvalues of stochastic blockmodel graphs and random graphs with low-rank edge probability matrices","volume":"84","author":"Athreya","year":"2022","journal-title":"Sankhya A"},{"key":"2024051307572568800_asad061-B7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s13171-015-0071-x","article-title":"A limit theorem for scaled eigenvectors of random dot product graphs","volume":"78","author":"Athreya","year":"2016","journal-title":"Sankhya A"},{"key":"2024051307572568800_asad061-B8","doi-asserted-by":"crossref","DOI":"10.1561\/0800000002","volume-title":"Large Dimensional Factor Analysis","author":"Bai","year":"2008"},{"key":"2024051307572568800_asad061-B9","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1111\/j.2044-8317.1995.tb01060.x","article-title":"Spearman and the origin and development of factor analysis","volume":"48","author":"Bartholomew","year":"1995","journal-title":"Br. J. Math. Statist. Psychol"},{"key":"2024051307572568800_asad061-B10","volume-title":"Confirmatory Factor Analysis for Applied Research","author":"Brown","year":"2015"},{"key":"2024051307572568800_asad061-B11","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1093\/biomet\/asy070","article-title":"Signal-plus-noise matrix models: eigenvector deviations and fluctuations","volume":"106","author":"Cape","year":"2019","journal-title":"Biometrika"},{"key":"2024051307572568800_asad061-B12","doi-asserted-by":"crossref","first-page":"2405","DOI":"10.1214\/18-AOS1752","article-title":"The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics","volume":"47","author":"Cape","year":"2019","journal-title":"Ann. Statist."},{"key":"2024051307572568800_asad061-B13","doi-asserted-by":"crossref","first-page":"1281","DOI":"10.2307\/1912275","article-title":"Arbitrage, factor structure, and mean-variance analysis on large asset markets","volume":"51","author":"Chamberlain","year":"1983","journal-title":"Econometrica"},{"key":"2024051307572568800_asad061-B14","author":"Chen","year":"2023"},{"key":"2024051307572568800_asad061-B15","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1561\/2200000079","article-title":"Spectral methods for data science: a statistical perspective","volume":"14","author":"Chen","year":"2021","journal-title":"Found. Trends Mach. Learn"},{"key":"2024051307572568800_asad061-B16","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1037\/1082-989X.4.3.272","article-title":"Evaluating the use of exploratory factor analysis in psychological research","volume":"4","author":"Fabrigar","year":"1999","journal-title":"Psychol. Meth"},{"key":"2024051307572568800_asad061-B17","first-page":"1","article-title":"An \u2113\u221e eigenvector perturbation bound and its application","volume":"18","author":"Fan","year":"2018","journal-title":"J. Mach. Learn. Res"},{"key":"2024051307572568800_asad061-B18","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1214\/20-STS785","article-title":"Robust high-dimensional factor models with applications to statistical machine learning","volume":"36","author":"Fan","year":"2021","journal-title":"Statist. Sci"},{"key":"2024051307572568800_asad061-B19","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0378-8733(83)90021-7","article-title":"Stochastic blockmodels: first steps","volume":"5","author":"Holland","year":"1983","journal-title":"Social Networks"},{"key":"2024051307572568800_asad061-B20","volume-title":"Factor Analysis: A Synthesis of Factorial Methods","author":"Holzinger","year":"1941"},{"key":"2024051307572568800_asad061-B21","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/BF02289233","article-title":"The varimax criterion for analytic rotation in factor analysis","volume":"23","author":"Kaiser","year":"1958","journal-title":"Psychometrika"},{"key":"2024051307572568800_asad061-B22","doi-asserted-by":"crossref","first-page":"016107","DOI":"10.1103\/PhysRevE.83.016107","article-title":"Stochastic blockmodels and community structure in networks","volume":"83","author":"Karrer","year":"2011","journal-title":"Phys. Rev. E"},{"key":"2024051307572568800_asad061-B23","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1214\/15-AOS1370","article-title":"A goodness-of-fit test for stochastic block models","volume":"44","author":"Lei","year":"2016","journal-title":"Ann. Statist."},{"key":"2024051307572568800_asad061-B24","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1214\/14-AOS1274","article-title":"Consistency of spectral clustering in stochastic block models","volume":"43","author":"Lei","year":"2015","journal-title":"Ann. Statist."},{"key":"2024051307572568800_asad061-B25","author":"R Development Core Team","year":"2024"},{"key":"2024051307572568800_asad061-B26","doi-asserted-by":"crossref","first-page":"1878","DOI":"10.1214\/11-AOS887","article-title":"Spectral clustering and the high-dimensional stochastic blockmodel","volume":"39","author":"Rohe","year":"2011","journal-title":"Ann. Statist."},{"key":"2024051307572568800_asad061-B27","doi-asserted-by":"crossref","first-page":"1037","DOI":"10.1093\/jrsssb\/qkad029","article-title":"Vintage factor analysis with varimax performs statistical inference","volume":"85","author":"Rohe","year":"2023","journal-title":"J. R. Statist. Soc. B"},{"key":"2024051307572568800_asad061-B28","doi-asserted-by":"crossref","first-page":"1446","DOI":"10.1111\/rssb.12509","article-title":"A statistical interpretation of spectral embedding: the generalised random dot product graph","volume":"84","author":"Rubin-Delanchy","year":"2022","journal-title":"J. R. Statist. Soc. B"},{"key":"2024051307572568800_asad061-B29","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.3150\/21-BEJ1376","article-title":"Asymptotically efficient estimators for stochastic blockmodels: the naive MLE, the rank-constrained MLE, and the spectral","volume":"28","author":"Tang","year":"2022","journal-title":"Bernoulli"},{"key":"2024051307572568800_asad061-B30","doi-asserted-by":"crossref","first-page":"2360","DOI":"10.1214\/17-AOS1623","article-title":"Limit theorems for eigenvectors of the normalized Laplacian for random graphs","volume":"46","author":"Tang","year":"2018","journal-title":"Ann. Statist."},{"key":"2024051307572568800_asad061-B31","doi-asserted-by":"crossref","DOI":"10.1037\/10018-000","volume-title":"The Vectors of Mind: Multiple-Factor Analysis for the Isolation of Primary Traits","author":"Thurstone","year":"1935"},{"key":"2024051307572568800_asad061-B32","volume-title":"Multiple-Factor Analysis: A Development and Expansion of \u2018The Vectors of Mind\u2019","author":"Thurstone","year":"1947"},{"key":"2024051307572568800_asad061-B33","doi-asserted-by":"crossref","first-page":"107","DOI":"10.2307\/2985729","article-title":"The origin and development of factor analysis","volume":"2","author":"Vincent","year":"1953","journal-title":"Appl. Statist"},{"key":"2024051307572568800_asad061-B34","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF01932678","article-title":"Perturbation bounds in connection with singular value decomposition","volume":"12","author":"Wedin","year":"1972","journal-title":"BIT Numer. Math"},{"key":"2024051307572568800_asad061-B35","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/BF01456804","article-title":"Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)","volume":"71","author":"Weyl","year":"1912","journal-title":"Math. Ann."},{"key":"2024051307572568800_asad061-B36","author":"Yan","year":"2021"},{"key":"2024051307572568800_asad061-B37","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1214\/21-AOS2074","article-title":"Heteroskedastic PCA: algorithm, optimality, and applications","volume":"50","author":"Zhang","year":"2022","journal-title":"Ann. Statist."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asad061\/56414952\/asad061.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/111\/2\/609\/57467522\/asad061.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/111\/2\/609\/57467522\/asad061.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,13]],"date-time":"2024-05-13T08:19:06Z","timestamp":1715588346000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/111\/2\/609\/7429398"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,20]]},"references-count":37,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,11,20]]},"published-print":{"date-parts":[[2024,5,13]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asad061","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2024,6,1]]},"published":{"date-parts":[[2023,11,20]]}}}