{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:25:20Z","timestamp":1787333120324,"version":"build-2736575974"},"reference-count":45,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"name":"DELPHI consortium"},{"DOI":"10.13039\/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["613.009.032"],"award-info":[{"award-number":["613.009.032"]}],"id":[{"id":"10.13039\/501100003246","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>We consider regularized least-squares problems of the form $\\min_{x} \\frac{1}{2}\\Vert Ax - b\\Vert_2^2 + \\mathcal{R}(Lx)$. Recently, Zheng et al. [ IEEE Access, 7 (2019), pp. 1404--1423], proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable $y$ and solves $\\min_{x,y} \\frac{1}{2}\\Vert Ax - b\\Vert_2^2 + \\frac{\\kappa}{2}\\Vert Lx - y\\Vert_2^2 + \\mathcal{R}(x)$. By minimizing out the variable $x$, we obtain an equivalent optimization problem $\\min_{y} \\frac{1}{2} \\Vert F_{\\kappa}y - g_{\\kappa}\\Vert_2^2+\\mathcal{R}(y)$. In our work, we view the SR3 method as a way to approximately solve the regularized problem. We analyze the conditioning of the relaxed problem in general and give an expression for the SVD of $F_{\\kappa}$ as a function of $\\kappa$. Furthermore, we relate the Pareto curve of the original problem to the relaxed problem, and we quantify the error incurred by relaxation in terms of $\\kappa$. Finally, we propose an efficient iterative method for solving the relaxed problem with inexact inner iterations. Numerical examples illustrate the approach.<\/jats:p>","DOI":"10.1137\/20m1348091","type":"journal-article","created":{"date-parts":[[2021,5,20]],"date-time":"2021-05-20T12:02:42Z","timestamp":1621512162000},"page":"S269-S292","source":"Crossref","is-referenced-by-count":2,"title":["Relaxed Regularization for Linear Inverse Problems"],"prefix":"10.1137","volume":"43","author":[{"given":"Nick","family":"Luiken","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8794-6426","authenticated-orcid":true,"given":"Tristan","family":"van Leeuwen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,5,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-019-00165-y"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/120899157"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/25\/12\/123010"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1887\/0750304359"},{"key":"atypb6","volume-title":"Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne","author":"Besson A.","year":"2019"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/18\/1\/201"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1561\/2200000016"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1023\/A:1026083619097"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1023\/A:1016656923184"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2005.862083"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20042"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-017-9911-z"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-017-1160-5"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01934412"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-018-0570-7"},{"key":"atypb17","first-page":"83","volume":"44","author":"Gazzola S.","year":"2015","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb18","volume-title":"Proceedings of the Workshop on Scientific Computing","author":"Golub G. H.","year":"1997"},{"key":"atypb19","volume-title":"Deep Learning","author":"Goodfellow I.","year":"2016"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/BF02219234"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/BF02149761"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719697"},{"key":"atypb23","first-page":"323","volume-title":"Numer. Algorithms","author":"Hansen P. C.","year":"2002"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718836"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1002\/nla.802"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-017-0430-x"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1190\/1.2841038"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-723-7"},{"key":"atypb29","first-page":"463","volume":"22","author":"Hochstenbach M.","year":"2010","journal-title":"J. Integral Equations"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-015-9990-x"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1137\/050645592"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899345960"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719284"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/18\/1\/315"},{"key":"atypb35","volume-title":"https:\/\/github.com\/bodono\/apg","author":"O'Donoghue B.","year":"2016"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1109\/ICIP.1994.413269"},{"key":"atypb37","first-page":"1458","volume-title":"Advances in Neural Information Processing Systems 24","author":"Schmidt M.","year":"2011"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/699\/1\/012010"},{"key":"atypb39","volume-title":"Proceedings of the ESA Earth Observation Summer School on Earth System Monitoring and Modeling","author":"Tamminen J.","year":"2012"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1137\/080714488"},{"key":"atypb41","volume-title":"Non-smooth Variable Projection, preprint, arXiv:1601.05011","author":"van Leeuwen T.","year":"2020"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1190\/1.3238367"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2018.2886528"},{"key":"atypb44","volume-title":"Principles of Magnetic Resonance Imaging: A Signal Processing Perspective","author":"Zhi-Pei L.","year":"2000"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-016-0271-0"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1348091","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:33:51Z","timestamp":1787330031000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1348091"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":45,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1348091"],"URL":"https:\/\/doi.org\/10.1137\/20m1348091","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}