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Through this universality, we introduce a notion of equivalence for estimation problems of this type and, in particular, show that a broad class of estimation tasks, including community detection, sparse submatrix detection and nonlinear spiked matrix models, are equivalent to spiked matrix models. As an application, we obtain a complete description of the performance of the least-squares (or \u201cbest rank one\u201d) estimator for any rank one matrix estimation problem.<\/jats:p>","DOI":"10.1214\/26-aos2639","type":"journal-article","created":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T08:18:26Z","timestamp":1787041106000},"page":"2031-2054","source":"Crossref","is-referenced-by-count":1,"title":["Pseudo-maximum likelihood theory for high-dimensional rank one inference"],"prefix":"10.1214","volume":"54","author":[{"given":"Curtis","family":"Grant","sequence":"first","affiliation":[{"name":"Department of Mathematics, Northwestern University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aukosh","family":"Jagannath","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Justin","family":"Ko","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"3","doi-asserted-by":"publisher","unstructured":"<small>Auffinger, A.<\/small> and <small>Chen, W.-K.<\/small> (2015). 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