{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:28:18Z","timestamp":1787318898199,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>The convergence rate is analyzed for the sparse reconstruction by separable approximation (SpaRSA) algorithm for minimizing a sum $f(\\mathbf{x})+\\psi(\\mathbf{x})$, where f is smooth and $\\psi$ is convex, but possibly nonsmooth. It is shown that if f is convex, then the error in the objective function at iteration k is bounded by $a\/k$ for some a independent of k. Moreover, if the objective function is strongly convex, then the convergence is R-linear. An improved version of the algorithm based on a cyclic version of the BB iteration and an adaptive line search is given. The performance of the algorithm is investigated using applications in the areas of signal processing and image reconstruction.<\/jats:p>","DOI":"10.1137\/090775063","type":"journal-article","created":{"date-parts":[[2011,2,15]],"date-time":"2011-02-15T21:36:19Z","timestamp":1297805779000},"page":"146-165","source":"Crossref","is-referenced-by-count":39,"title":["Gradient-Based Methods for Sparse Recovery"],"prefix":"10.1137","volume":"4","author":[{"given":"William W.","family":"Hager","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dzung T.","family":"Phan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hongchao","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,2,15]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2010.2047910"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/8.1.141"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/090756855"},{"key":"R5","unstructured":"J. Bioucas-Dias and M. Figueiredo,\n                      TwIST: Two-Step Iterative Shrinkage\/Thresholding Algorithm for Linear Inverse Problems\n                      , http:\/\/www.lx.it.pt\/ bioucas\/TwIST\/TwIST.htm."},{"key":"R6","doi-asserted-by":"crossref","unstructured":"J. Bioucas-Dias, M. Figueiredo, and J. P. Oliveira,\n                      Total variation-based image deconvolution: A majorization-minimization approach\n                      , in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Vol. 2, 2006, pp. 861\u2013864.","DOI":"10.1109\/ICASSP.2006.1660479"},{"key":"R7","unstructured":"R. Broughton, I. Coope, P. Renaud, and R. Tappenden,\n                      A Barzilai-Borwein $\\ell_1$-Regularized Least Squares Algorithm for Compressed Sensing\n                      , Technical report, 2009; available online from http:\/\/arxiv.org\/abs\/0907.3340 http:\/\/arxiv.org\/abs\/0907.3340."},{"key":"R8","doi-asserted-by":"crossref","unstructured":"E. J. Cand\u00e8s and J. Romberg,\n                      Practical signal recovery from random projections\n                      , in Proceedings of the SPIE Wavelet Applications in Signal and Image Processing Conference XI, Vol. 5914, 2005.","DOI":"10.1117\/12.600722"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1109\/83.661182"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596304010"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1080\/02331930310001611547"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drl006"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20042"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1109\/JSTSP.2007.910281"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2003.814255"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/S003614299427315X"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/0723046"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/050635225"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/070698920"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1109\/JSTSP.2007.910971"},{"key":"R21","unstructured":"Z. Lu and Y. Zhang,\n                      An Augmented Lagrangian Approach for Sparse Principal Component Analysis\n                      , Technical report, Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada, 2009."},{"key":"R22","unstructured":"Y. Nesterov,\n                      Gradient Methods for Minimizing Composite Objective Function\n                      , CORE Discussion Papers 2007\/76, Center for Operations Research and Econometrics (CORE), Universit\u00e9 catholique de Louvain, Louvain-la-Neuve, Belgium, 2007."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013708715892"},{"key":"R24","unstructured":"R. T. Rockafellar,\n                      Convex Analysis\n                      , Princeton University Press, Princeton, NJ, 1970."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1145\/1462173.1462178"},{"key":"R26","unstructured":"L. Vandenberghe,\n                      Gradient Methods for Nonsmooth Problems\n                      , http:\/\/www.ee.ucla.edu\/ vandenbe\/ee236c.html http:\/\/www.ee.ucla.edu\/ vandenbe\/ee236c.html."},{"key":"R27","doi-asserted-by":"crossref","unstructured":"C. Vonesch and M. Unser,\n                      Fast iterative thresholding algorithm for wavelet-regularized deconvolution\n                      , in Proceedings of the SPIE Optics and Photonics 2007 Conference on Mathematical Methods: Wavelet XII, Vol. 6701, San Diego, CA, 2007, pp. 1\u20135.","DOI":"10.1117\/12.733532"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2009.2016892"}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090775063","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:43:16Z","timestamp":1787316196000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090775063"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1137\/090775063"],"URL":"https:\/\/doi.org\/10.1137\/090775063","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}