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A simple thresholding estimator of the singular values achieves the rate (q\u2227p)(pq)1\/4, which turns out to be optimal up to a logarithmic multiplicative term. The tight minimax rate is achieved by a more involved polynomial approximation method. This allows us to build estimators for a class of effective rank indices. As a byproduct, we also characterize the minimax rate for estimating the sequence of singular values of a matrix.<\/jats:p>","DOI":"10.1214\/24-aos2374","type":"journal-article","created":{"date-parts":[[2024,10,4]],"date-time":"2024-10-04T01:47:46Z","timestamp":1728006466000},"page":"1334-1359","source":"Crossref","is-referenced-by-count":1,"title":["Optimal estimation of Schatten norms of a rectangular matrix"],"prefix":"10.1214","volume":"52","author":[{"given":"Sol\u00e8ne","family":"Th\u00e9paut","sequence":"first","affiliation":[{"name":"Safran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicolas","family":"Verzelen","sequence":"additional","affiliation":[{"name":"MISTEA, INRAE, Universit\u00e9 de Montpellier"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"32","doi-asserted-by":"publisher","unstructured":"Vershynin, R.<\/small> (2018). 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