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The effectiveness of the proposed algorithm is demonstrated in a suite of problems in high-dimensional statistics, including singular value decomposition (SVD) under heteroskedastic noise, Poisson PCA, and SVD for heteroskedastic and incomplete data.<\/jats:p>","DOI":"10.1214\/21-aos2074","type":"journal-article","created":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T01:22:28Z","timestamp":1645060948000},"page":"53-80","source":"Crossref","is-referenced-by-count":51,"title":["Heteroskedastic PCA: Algorithm, optimality, and applications"],"prefix":"10.1214","volume":"50","author":[{"given":"Anru R.","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Wisconsin-Madison"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"T. Tony","family":"Cai","sequence":"additional","affiliation":[{"name":"Department of Statistics, The Wharton School, University of Pennsylvania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yihong","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, Yale University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"28","doi-asserted-by":"crossref","unstructured":"<small>Hotelling, H.<\/small> (1936). Relations between two sets of variates. <i>Biometrika<\/i> <b>28<\/b> 321\u2013377.","DOI":"10.1093\/biomet\/28.3-4.321"},{"key":"12","doi-asserted-by":"publisher","unstructured":"<small>Cand\u00e8s, E. J.<\/small>, <small>Sing-Long, C. A.<\/small> and <small>Trzasko, J. D.<\/small> (2013). Unbiased risk estimates for singular value thresholding and spectral estimators. <i>IEEE Trans. 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Available at <a href=\"https:\/\/arxiv.org\/abs\/arXiv:2008.12434\">arXiv:2008.12434<\/a>."},{"key":"10","doi-asserted-by":"publisher","unstructured":"<small>CAI, T. T.<\/small> and <small>ZHANG, A.<\/small> (2018). Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics. <i>Ann. Statist.<\/i> <b>46<\/b> 60\u201389.","DOI":"10.1214\/17-AOS1541"},{"key":"16","doi-asserted-by":"crossref","unstructured":"<small>COLLINS, M.<\/small>, <small>DASGUPTA, S.<\/small> and <small>SCHAPIRE, R. E.<\/small> (2002). A generalization of principal components analysis to the exponential family. In <i>Advances in Neural Information Processing Systems<\/i> 617\u2013624.","DOI":"10.7551\/mitpress\/1120.003.0084"},{"key":"18","doi-asserted-by":"publisher","unstructured":"<small>DE LATHAUWER, L.<\/small>, <small>DE MOOR, B.<\/small> and <small>VANDEWALLE, J.<\/small> (2000). On the best rank-1 and rank-<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" alttext=\"$({R_{1}},{R_{2}},\\dots ,{R_{N}})$\"><mo fence=\"true\" stretchy=\"false\">(<\/mo><msub><mrow><mi mathvariant=\"italic\">R<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo>,<\/mo><msub><mrow><mi mathvariant=\"italic\">R<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mrow><mi mathvariant=\"italic\">R<\/mi><\/mrow><mrow><mi mathvariant=\"italic\">N<\/mi><\/mrow><\/msub><mo fence=\"true\" stretchy=\"false\">)<\/mo><\/math> approximation of higher-order tensors. <i>SIAM J. Matrix Anal. Appl.<\/i> <b>21<\/b> 1324\u20131342.","DOI":"10.1137\/S0895479898346995"},{"key":"21","unstructured":"<small>FLORESCU, L.<\/small> and <small>PERKINS, W.<\/small> (2016). Spectral thresholds in the bipartite stochastic block model. In <i>Conference on Learning Theory<\/i> 943\u2013959."},{"key":"22","doi-asserted-by":"publisher","unstructured":"<small>FORTUNATO, S.<\/small> (2010). Community detection in graphs. <i>Phys. Rep.<\/i> <b>486<\/b> 75\u2013174.","DOI":"10.1016\/j.physrep.2009.11.002"},{"key":"23","doi-asserted-by":"publisher","unstructured":"<small>GAVISH, M.<\/small> and <small>DONOHO, D. L.<\/small> (2017). Optimal shrinkage of singular values. <i>IEEE Trans. Inf. Theory<\/i> <b>63<\/b> 2137\u20132152.","DOI":"10.1109\/TIT.2017.2653801"},{"key":"24","doi-asserted-by":"publisher","unstructured":"<small>GHOSH, J.<\/small> and <small>DUNSON, D. B.<\/small> (2009). Default prior distributions and efficient posterior computation in Bayesian factor analysis. <i>J. Comput. Graph. Statist.<\/i> <b>18<\/b> 306\u2013320.","DOI":"10.1198\/jcgs.2009.07145"},{"key":"25","doi-asserted-by":"crossref","unstructured":"<small>HONG, D.<\/small>, <small>BALZANO, L.<\/small> and <small>FESSLER, J. A.<\/small> (2016). 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Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. <i>Statist. Sinica<\/i> <b>17<\/b> 1617\u20131642."},{"key":"56","doi-asserted-by":"publisher","unstructured":"<small>Wang, W.<\/small> and <small>Fan, J.<\/small> (2017). Asymptotics of empirical eigenstructure for high dimensional spiked covariance. <i>Ann. Statist.<\/i> <b>45<\/b> 1342\u20131374.","DOI":"10.1214\/16-AOS1487"},{"key":"57","doi-asserted-by":"publisher","unstructured":"<small>Wedin, P.<\/small> (1972). Perturbation bounds in connection with singular value decomposition. <i>BIT<\/i> <b>12<\/b> 99\u2013111.","DOI":"10.1007\/bf01932678"},{"key":"59","doi-asserted-by":"publisher","unstructured":"<small>Yu, Y.<\/small>, <small>Wang, T.<\/small> and <small>Samworth, R. J.<\/small> (2015). 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