{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,24]],"date-time":"2026-10-24T03:41:47Z","timestamp":1792813307100,"version":"4.5.0"},"reference-count":61,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2024,2,3]],"date-time":"2024-02-03T00:00:00Z","timestamp":1706918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/501100002920","name":"RGC","doi-asserted-by":"publisher","award":["26302019"],"award-info":[{"award-number":["26302019"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002920","name":"RGC","doi-asserted-by":"publisher","award":["16303320"],"award-info":[{"award-number":["16303320"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006445","name":"NSF","doi-asserted-by":"publisher","award":["DMS-1662139"],"award-info":[{"award-number":["DMS-1662139"]}],"id":[{"id":"10.13039\/100006445","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006445","name":"NSF","doi-asserted-by":"publisher","award":["DMS-1712591"],"award-info":[{"award-number":["DMS-1712591"]}],"id":[{"id":"10.13039\/100006445","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006445","name":"NSF","doi-asserted-by":"publisher","award":["DMS-2210833"],"award-info":[{"award-number":["DMS-2210833"]}],"id":[{"id":"10.13039\/100006445","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006445","name":"NSF","doi-asserted-by":"publisher","award":["N00014-22-1-2340"],"award-info":[{"award-number":["N00014-22-1-2340"]}],"id":[{"id":"10.13039\/100006445","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006445","name":"NSF","doi-asserted-by":"publisher","award":["DMS-1803241"],"award-info":[{"award-number":["DMS-1803241"]}],"id":[{"id":"10.13039\/100006445","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,7,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This paper introduces a general framework of Semi-parametric TEnsor Factor Analysis (STEFA) that focuses on the methodology and theory of low-rank tensor decomposition with auxiliary covariates. Semi-parametric TEnsor Factor Analysis models extend tensor factor models by incorporating auxiliary covariates in the loading matrices. We propose an algorithm of iteratively projected singular value decomposition (IP-SVD) for the semi-parametric estimation. It iteratively projects tensor data onto the linear space spanned by the basis functions of covariates and applies singular value decomposition on matricized tensors over each mode. We establish the convergence rates of the loading matrices and the core tensor factor. The theoretical results only require a sub-exponential noise distribution, which is weaker than the assumption of sub-Gaussian tail of noise in the literature. Compared with the Tucker decomposition, IP-SVD yields more accurate estimators with a faster convergence rate. Besides estimation, we propose several prediction methods with new covariates based on the STEFA model. On both synthetic and real tensor data, we demonstrate the efficacy of the STEFA model and the IP-SVD algorithm on both the estimation and prediction tasks.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkae001","type":"journal-article","created":{"date-parts":[[2024,2,3]],"date-time":"2024-02-03T19:22:39Z","timestamp":1706988159000},"page":"793-823","source":"Crossref","is-referenced-by-count":21,"title":["Semi-parametric tensor factor analysis by iteratively projected singular value decomposition"],"prefix":"10.1093","volume":"86","author":[{"given":"Elynn Y","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Technology, Operations and Statistics, New York University , New York, NY , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dong","family":"Xia","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hong Kong University of Science and Technology , Clear Water Bay, Kowloon, Hong Kong , China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chencheng","family":"Cai","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Washington State University , Pullman, WA , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3250-7677","authenticated-orcid":false,"given":"Jianqing","family":"Fan","sequence":"additional","affiliation":[{"name":"School of Data Science, Fudan University , Shanghai , China"},{"name":"Department of Operations Research and Financial Engineering, Princeton University , Princeton, NJ , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,2,3]]},"reference":[{"key":"2024071202434093000_qkae001-B1","author":"Acar","year":"2011"},{"issue":"3","key":"2024071202434093000_qkae001-B2","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.3982\/ECTA8968","article-title":"Eigenvalue ratio test for the number of factors","volume":"81","author":"Ahn","year":"2013","journal-title":"Econometrica"},{"key":"2024071202434093000_qkae001-B3","author":"Allen","year":"2012"},{"key":"2024071202434093000_qkae001-B4","author":"Allen","year":"2012"},{"key":"2024071202434093000_qkae001-B5","volume-title":"An introduction to the multivariate statistical analysis","author":"Anderson","year":"2003","edition":"3rd ed."},{"key":"2024071202434093000_qkae001-B6","volume-title":"proceedings of the third berkeley symposium on mathematical statistics and probability","author":"Anderson","year":"1956"},{"key":"2024071202434093000_qkae001-B7","author":"Bahadori","year":"2014"},{"issue":"1","key":"2024071202434093000_qkae001-B8","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1111\/1468-0262.00392","article-title":"Inferential theory for factor models of large dimensions","volume":"71","author":"Bai","year":"2003","journal-title":"Econometrica"},{"issue":"1","key":"2024071202434093000_qkae001-B9","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1214\/11-AOS966","article-title":"Statistical analysis of factor models of high dimension","volume":"40","author":"Bai","year":"2012","journal-title":"The Annals of Statistics"},{"issue":"1","key":"2024071202434093000_qkae001-B10","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1111\/1468-0262.00273","article-title":"Determining the number of factors in approximate factor models","volume":"70","author":"Bai","year":"2002","journal-title":"Econometrica"},{"key":"2024071202434093000_qkae001-B11","author":"Baskaran","year":"2017"},{"key":"2024071202434093000_qkae001-B12","author":"Cai","year":"2019"},{"issue":"5","key":"2024071202434093000_qkae001-B13","doi-asserted-by":"crossref","first-page":"2098","DOI":"10.1214\/009053606000000641","article-title":"Discussion: Conditional growth charts","volume":"34","author":"Carroll","year":"2006","journal-title":"The Annals of Statistics"},{"issue":"30","key":"2024071202434093000_qkae001-B14","doi-asserted-by":"crossref","DOI":"10.1073\/pnas.2026290118","article-title":"Observational evidence that cloud feedback amplifies global warming","volume":"118","author":"Ceppi","year":"2021","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"542","key":"2024071202434093000_qkae001-B15","doi-asserted-by":"crossref","first-page":"1038","DOI":"10.1080\/01621459.2021.1970569","article-title":"Statistical inference for high-dimensional matrix-variate factor models","volume":"118","author":"Chen","year":"2023","journal-title":"Journal of the American Statistical Association"},{"issue":"530","key":"2024071202434093000_qkae001-B16","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1080\/01621459.2019.1584899","article-title":"Constrained factor models for high-dimensional matrix-variate time series","volume":"115","author":"Chen","year":"2019","journal-title":"Journal of the American Statistical Association"},{"key":"2024071202434093000_qkae001-B17","first-page":"1","author":"Chen","year":"2024"},{"key":"2024071202434093000_qkae001-B18","author":"Chen","year":"2020"},{"issue":"537","key":"2024071202434093000_qkae001-B19","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1080\/01621459.2021.1912757","article-title":"Factor models for high-dimensional tensor time series","volume":"117","author":"Chen","year":"2022","journal-title":"Journal of the American Statistical Association"},{"key":"2024071202434093000_qkae001-B20","doi-asserted-by":"crossref","first-page":"5549","DOI":"10.1016\/S1573-4412(07)06076-X","article-title":"Large sample sieve estimation of semi-nonparametric models","volume":"6","author":"Chen","year":"2007","journal-title":"Handbook of Econometrics"},{"issue":"5","key":"2024071202434093000_qkae001-B21","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1162\/jocn.2009.21062","article-title":"Good vibrations: Cross-frequency coupling in the human nucleus accumbens during reward processing","volume":"21","author":"Cohen","year":"2009","journal-title":"Journal of Cognitive Neuroscience"},{"issue":"2","key":"2024071202434093000_qkae001-B22","doi-asserted-by":"crossref","first-page":"713","DOI":"10.3982\/ECTA7432","article-title":"Efficient semiparametric estimation of the Fama\u2013French model and extensions","volume":"80","author":"Connor","year":"2012","journal-title":"Econometrica"},{"issue":"5","key":"2024071202434093000_qkae001-B23","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1016\/j.jempfin.2006.10.001","article-title":"Semiparametric estimation of a characteristic-based factor model of common stock returns","volume":"14","author":"Connor","year":"2007","journal-title":"Journal of Empirical Finance"},{"key":"2024071202434093000_qkae001-B24","author":"De Almeida","year":"2014"},{"issue":"4","key":"2024071202434093000_qkae001-B25","doi-asserted-by":"crossref","first-page":"1324","DOI":"10.1137\/S0895479898346995","article-title":"On the best rank-1 and rank-(r1,r2,\u2026,rn) approximation of higher-order tensors","volume":"21","author":"De Lathauwer","year":"2000","journal-title":"SIAM Journal on Matrix Analysis and Applications"},{"issue":"4","key":"2024071202434093000_qkae001-B26","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1137\/S0895479896305696","article-title":"A multilinear singular value decomposition","volume":"21","author":"De Lathauwer","year":"2000","journal-title":"SIAM Journal on Matrix Analysis and Applications"},{"issue":"3","key":"2024071202434093000_qkae001-B27","doi-asserted-by":"crossref","first-page":"968","DOI":"10.1016\/j.neuroimage.2006.01.021","article-title":"An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest","volume":"31","author":"Desikan","year":"2006","journal-title":"Neuroimage"},{"key":"2024071202434093000_qkae001-B28","doi-asserted-by":"crossref","DOI":"10.1201\/9780429096280","volume-title":"Statistical foundations of data science","author":"Fan","year":"2020"},{"issue":"1","key":"2024071202434093000_qkae001-B29","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1214\/15-AOS1364","article-title":"Projected principal component analysis in factor models","volume":"44","author":"Fan","year":"2016","journal-title":"Annals of Statistics"},{"key":"2024071202434093000_qkae001-B30","author":"Fan","year":"2020"},{"key":"2024071202434093000_qkae001-B31","author":"Han","year":"2020"},{"issue":"1","key":"2024071202434093000_qkae001-B32","doi-asserted-by":"crossref","first-page":"1726","DOI":"10.1214\/22-EJS1991","article-title":"Rank determination in tensor factor model","volume":"16","author":"Han","year":"2022","journal-title":"Electronic Journal of Statistics"},{"issue":"1","key":"2024071202434093000_qkae001-B33","first-page":"2989","article-title":"Sparse tensor additive regression","volume":"22","author":"Hao","year":"2021","journal-title":"The Journal of Machine Learning Research"},{"issue":"1","key":"2024071202434093000_qkae001-B34","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1080\/10618600.2021.1978471","article-title":"Generalized tensor decomposition with features on multiple modes","volume":"31","author":"Hu","year":"2022","journal-title":"Journal of Computational and Graphical Statistics"},{"issue":"488","key":"2024071202434093000_qkae001-B35","doi-asserted-by":"crossref","first-page":"1609","DOI":"10.1198\/jasa.2009.tm08024","article-title":"The analysis of two-way functional data using two-way regularized singular value decompositions","volume":"104","author":"Huang","year":"2009","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"2024071202434093000_qkae001-B36","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1137\/07070111X","article-title":"Tensor decompositions and applications","volume":"51","author":"Kolda","year":"2009","journal-title":"SIAM Review"},{"issue":"2","key":"2024071202434093000_qkae001-B37","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1214\/12-AOS970","article-title":"Factor modeling for high-dimensional time series: Inference for the number of factors","volume":"40","author":"Lam","year":"2012","journal-title":"The Annals of Statistics"},{"key":"2024071202434093000_qkae001-B38","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.jmva.2015.02.016","article-title":"Supervised singular value decomposition and its asymptotic properties","volume":"146","author":"Li","year":"2016","journal-title":"Journal of Multivariate Analysis"},{"key":"2024071202434093000_qkae001-B39","author":"Lozano","year":"2009"},{"issue":"525","key":"2024071202434093000_qkae001-B40","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1080\/01621459.2017.1389740","article-title":"Matrix completion with covariate information","volume":"114","author":"Mao","year":"2019","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"2024071202434093000_qkae001-B41","doi-asserted-by":"crossref","first-page":"1554","DOI":"10.1214\/18-AOS1725","article-title":"Convex regularization for high-dimensional multiresponse tensor regression","volume":"47","author":"Raskutti","year":"2019","journal-title":"The Annals of Statistics"},{"key":"2024071202434093000_qkae001-B42","author":"Richard","year":"2014"},{"issue":"2","key":"2024071202434093000_qkae001-B43","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1159\/000368279","article-title":"The nucleus accumbens: A comprehensive review","volume":"93","author":"Salgado","year":"2015","journal-title":"Stereotactic and Functional Neurosurgery"},{"issue":"7","key":"2024071202434093000_qkae001-B44","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1038\/ng1589","article-title":"An integrative genomics approach to infer causal associations between gene expression and disease","volume":"37","author":"Schadt","year":"2005","journal-title":"Nature Genetics"},{"issue":"1","key":"2024071202434093000_qkae001-B45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/aos\/1033066196","article-title":"Smoothed functional principal components analysis by choice of norm","volume":"24","author":"Silverman","year":"1996","journal-title":"The Annals of Statistics"},{"issue":"1","key":"2024071202434093000_qkae001-B46","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3278607","article-title":"Tensor completion algorithms in big data analytics","volume":"13","author":"Song","year":"2019","journal-title":"ACM Transactions on Knowledge Discovery from Data (TKDD)"},{"issue":"1","key":"2024071202434093000_qkae001-B47","first-page":"4908","article-title":"Store: Sparse tensor response regression and neuroimaging analysis","volume":"18","author":"Sun","year":"2017","journal-title":"The Journal of Machine Learning Research"},{"issue":"3","key":"2024071202434093000_qkae001-B48","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1111\/rssb.12190","article-title":"Provable sparse tensor decomposition","volume":"79","author":"Sun","year":"2017","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"2024071202434093000_qkae001-B49","volume-title":"Introduction to nonparametric estimation","author":"Tsybakov","year":"2008"},{"issue":"1","key":"2024071202434093000_qkae001-B50","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.jeconom.2018.09.013","article-title":"Factor models for matrix-valued high-dimensional time series","volume":"208","author":"Wang","year":"2019","journal-title":"Journal of Econometrics"},{"issue":"154","key":"2024071202434093000_qkae001-B51","first-page":"1","article-title":"Learning from binary multiway data: Probabilistic tensor decomposition and its statistical optimality","volume":"21","author":"Wang","year":"2020","journal-title":"The Journal of Machine Learning Research"},{"key":"2024071202434093000_qkae001-B52","author":"Wang","year":"2017"},{"issue":"1","key":"2024071202434093000_qkae001-B53","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1080\/10618600.2016.1157483","article-title":"Regularized principal component analysis for spatial data","volume":"26","author":"Wang","year":"2017","journal-title":"Journal of Computational and Graphical Statistics"},{"issue":"1","key":"2024071202434093000_qkae001-B54","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1523\/JNEUROSCI.3489-08.2009","article-title":"Relating structure to function: Heschl\u2019s gyrus and acoustic processing","volume":"29","author":"Warrier","year":"2009","journal-title":"Journal of Neuroscience"},{"issue":"1","key":"2024071202434093000_qkae001-B55","first-page":"2206","article-title":"The sup-norm perturbation of HOSVD and low rank tensor denoising","volume":"20","author":"Xia","year":"2019","journal-title":"Journal of Machine Learning Research"},{"key":"2024071202434093000_qkae001-B56","author":"Xu","year":"2019"},{"issue":"2","key":"2024071202434093000_qkae001-B57","doi-asserted-by":"crossref","first-page":"936","DOI":"10.1214\/18-AOS1694","article-title":"Cross: Efficient low-rank tensor completion","volume":"47","author":"Zhang","year":"2019","journal-title":"The Annals of Statistics"},{"issue":"528","key":"2024071202434093000_qkae001-B58","doi-asserted-by":"crossref","first-page":"1708","DOI":"10.1080\/01621459.2018.1527227","article-title":"Optimal sparse singular value decomposition for high-dimensional high-order data","volume":"114","author":"Zhang","year":"2019","journal-title":"Journal of the American Statistical Association"},{"issue":"11","key":"2024071202434093000_qkae001-B59","doi-asserted-by":"crossref","first-page":"7311","DOI":"10.1109\/TIT.2018.2841377","article-title":"Tensor SVD: Statistical and computational limits","volume":"64","author":"Zhang","year":"2018","journal-title":"IEEE Transactions on Information Theory"},{"issue":"541","key":"2024071202434093000_qkae001-B60","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1080\/01621459.2021.1938082","article-title":"Partially observed dynamic tensor response regression","volume":"118","author":"Zhou","year":"2021","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"2024071202434093000_qkae001-B61","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1080\/10618600.2013.827986","article-title":"Principal component analysis of two-dimensional functional data","volume":"23","author":"Zhou","year":"2014","journal-title":"Journal of Computational and Graphical Statistics"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/86\/3\/793\/58504454\/qkae001.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/86\/3\/793\/58504454\/qkae001.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T02:45:17Z","timestamp":1720752317000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/86\/3\/793\/7600270"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,3]]},"references-count":61,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,2,3]]},"published-print":{"date-parts":[[2024,7,12]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkae001","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2024,7]]},"published":{"date-parts":[[2024,2,3]]}}}