{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:00:49Z","timestamp":1760241649904,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,7,24]],"date-time":"2018-07-24T00:00:00Z","timestamp":1532390400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for the closest rank-r matrix X to M, where the distance is measured in the Frobenius norm. A key element in the solution of this matrix nearness problem consists of the use of a constrained gradient system of matrix differential equations. The obtained results, compared to those obtained by different approaches show that the method has a correct behaviour and is competitive with the ones available in the literature.<\/jats:p>","DOI":"10.3390\/axioms7030051","type":"journal-article","created":{"date-parts":[[2018,7,24]],"date-time":"2018-07-24T11:00:07Z","timestamp":1532430007000},"page":"51","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Gradient System for Low Rank Matrix Completion"],"prefix":"10.3390","volume":"7","author":[{"given":"Carmela","family":"Scalone","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria e Scienze dell\u2019Informazione e Matematica (DISIM), Universit\u00e0 dell\u2019Aquila, via Vetoio 1, 67100 L\u2019Aquila, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicola","family":"Guglielmi","sequence":"additional","affiliation":[{"name":"Section of Mathematics, Gran Sasso Science Institute, via Crispi 7, 67100 L\u2019Aquila, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1007\/s10208-009-9045-5","article-title":"Exact Matrix Completion via Convex Optimization","volume":"9","author":"Recht","year":"2009","journal-title":"Found. 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