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Optim."],"published-print":{"date-parts":[[2026,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this paper we derive sufficient conditions for the convergence of two popular alternating minimization algorithms for dictionary learning, the method of optimal directions (MOD) and online dictionary learning (ODL), which can also be thought of as approximative K-SVD. We show for a generating dictionary with [Formula: see text] atoms, that given enough training signals in each iteration and a well-behaved initialization that is either within distance at most [Formula: see text] to the generating dictionary or has a special structure, ensuring that each atom of the initialization only points to one generating atom, both algorithms will converge with geometric convergence rate to the generating dictionary. This is done even for signal models with nonuniform distributions on the supports of the sparse coefficients. These allow the appearance frequency of the dictionary atoms to vary heavily and thus model real data more closely.<\/jats:p>","DOI":"10.1137\/23m1575469","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T08:20:24Z","timestamp":1772094024000},"page":"320-349","source":"Crossref","is-referenced-by-count":0,"title":["Convergence Regions of Alternating Minimization Algorithms for Dictionary Learning"],"prefix":"10.1137","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0434-8982","authenticated-orcid":true,"given":"Simon","family":"Ruetz","sequence":"first","affiliation":[{"name":"Department of Mathematics, Universit\u00e4t Innsbruck, Innsbruck, 6020 Austria."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4873-5570","authenticated-orcid":true,"given":"Karin","family":"Schnass","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Universit\u00e4t Innsbruck, Innsbruck, 6020 Austria."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,2,26]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/140979861"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2016.2614684"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2006.881199"},{"key":"ref4","volume-title":"COLT 2015","author":"Arora S.","year":"2015"},{"key":"ref5","volume-title":"COLT 2014","author":"Arora S.","year":"2014"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1145\/2746539.2746605"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2005.862083"},{"key":"ref8","volume-title":"Advances in Neural Information Processing Systems 30 (NIPS 2017)","author":"Chatterji N.","year":"2017"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1155\/2020\/1657381"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2006.871582"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1214\/009053604000000067"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1109\/ICASSP.1999.760624"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1038\/381607a0"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2015.2472522"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2010.2048466"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2006.03.036"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1162\/089976603762552951"},{"key":"ref18","unstructured":"Y. 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