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Optim."],"published-print":{"date-parts":[[2016,1]]},"abstract":"<jats:p>We use a model LASSO problem to analyze the convergence behavior of the ISTA and FISTA iterations, showing that both iterations satisfy local linear convergence rate bound when close enough to the solution. Using the observation that FISTA is an accelerated ISTA process, and a spectral analysis of the associated matrix operators, we show that FISTA's convergence rate can slow down as it proceeds, eventually becoming slower than ISTA. This observation leads to a proposed heuristic algorithm to take an ISTA step if it shows more progress compared to FISTA, as measured by the decrease in the objective function. We illustrate the results with some synthetic numerical examples.<\/jats:p>","DOI":"10.1137\/151004549","type":"journal-article","created":{"date-parts":[[2016,2,2]],"date-time":"2016-02-02T11:57:39Z","timestamp":1454414259000},"page":"313-336","source":"Crossref","is-referenced-by-count":53,"title":["Local Linear Convergence of ISTA and FISTA on the LASSO Problem"],"prefix":"10.1137","volume":"26","author":[{"given":"Shaozhe","family":"Tao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Boley","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuzhong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,1,28]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"A. Ben-Tal and A. Nemirovski,\n                      Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications\n                      , MPS\/SIAM Ser. Optim., SIAM, Philadelphia, 2001.","DOI":"10.1137\/1.9780898718829"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/120878951"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s00041-008-9041-1"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1214\/08-AOS653"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596304010"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20042"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1214\/009053604000000067"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"L. Fuchs,\n                      Recovery of exact sparse representations in the presence of bounded noise\n                      , IEEE Trans. Inform. 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Moulin,\n                      A Lyapunov analysis of FISTA with local linear convergence for sparse optimization\n                      , preprint, arXiv:1502.02281, 2015."},{"key":"atypb14","first-page":"1970","volume":"27","author":"Liang J.","year":"2014","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"atypb15","first-page":"372","volume":"27","author":"Nesterov Y.","year":"1983","journal-title":"Soviet Math. Dokl."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-013-9150-3"},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"J. M. Ortega,\n                      Numerical Analysis: A Second Course\n                      , SIAM, Philadelphia 1990.","DOI":"10.1137\/1.9781611971323"},{"key":"atypb18","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1080\/10618600.2000.10474883","volume":"9","author":"Osborne M. R.","year":"1999","journal-title":"J. Comput. Graph. Statist."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1561\/2400000003"},{"key":"atypb20","unstructured":"S. Tao, D. Boley, and S. Zhang,\n                      Convergence of common proximal methods for $\\ell_1$ regularized least squares\n                      , in Proceedings of the International Joint Conference on Artificial Intelligence, 2015."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1190\/1.1440921"},{"key":"atypb22","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"Tibshirani R.","year":"1996","journal-title":"J. R. Statist. Soc. Ser. B (Methodological)"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1214\/13-EJS815"},{"key":"atypb24","unstructured":"H. Zhang, W. Yin, and L. Cheng,\n                      Necessary and sufficient conditions of solution uniqueness in $\\ell_1$ minimization\n                      , CoRR, abs\/1209.0652 (2012)."}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/151004549","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:52:09Z","timestamp":1787338329000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/151004549"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1]]},"references-count":24,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2016,1]]}},"alternative-id":["10.1137\/151004549"],"URL":"https:\/\/doi.org\/10.1137\/151004549","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1]]}}}