{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T15:26:41Z","timestamp":1761060401539,"version":"3.37.3"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2017,8,9]],"date-time":"2017-08-09T00:00:00Z","timestamp":1502236800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100002790","name":"Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["611496"],"award-info":[{"award-number":["611496"]}],"id":[{"id":"10.13039\/501100002790","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61373087","61472257"],"award-info":[{"award-number":["61373087","61472257"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Guangdong, China","award":["2015A030313550","2015A030313557"],"award-info":[{"award-number":["2015A030313550","2015A030313557"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2017,12]]},"DOI":"10.1007\/s10589-017-9933-6","type":"journal-article","created":{"date-parts":[[2017,8,9]],"date-time":"2017-08-09T08:25:28Z","timestamp":1502267128000},"page":"619-642","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["$$\\ell _p$$ \u2113 p Regularized low-rank approximation via iterative reweighted singular value minimization"],"prefix":"10.1007","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3277-7853","authenticated-orcid":false,"given":"Zhaosong","family":"Lu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Lu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2017,8,9]]},"reference":[{"key":"9933_CR1","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1093\/imanum\/8.1.141","volume":"8","author":"J Barzilai","year":"1988","unstructured":"Barzilai, J., Borwein, J.M.: Two point step size gradient methods. IMA J. Numer. Anal. 8, 141\u2013148 (1988)","journal-title":"IMA J. Numer. Anal."},{"key":"9933_CR2","doi-asserted-by":"crossref","unstructured":"Bertalm\u00edo, M., Sapiro, G., Caselles, V., Ballester C.: Image inpainting. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH) (2000)","DOI":"10.1145\/344779.344972"},{"key":"9933_CR3","unstructured":"Bian, W., Chen, X.: Worst-case complexity of smoothing quadratic regularization methods for non-Lipschitzian optimization, SIAM J. Optim. 23, 1718\u20131741 (2013)"},{"key":"9933_CR4","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1137\/S1052623497330963","volume":"4","author":"EG Birgin","year":"2000","unstructured":"Birgin, E.G., Mart\u00ednez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 4, 1196\u20131211 (2000)","journal-title":"SIAM J. Optim."},{"key":"9933_CR5","doi-asserted-by":"crossref","first-page":"1956","DOI":"10.1137\/080738970","volume":"20","author":"J-F Cai","year":"2010","unstructured":"Cai, J.-F., Cand\u00e9s, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 1956\u20131982 (2010)","journal-title":"SIAM J. Optim."},{"key":"9933_CR6","unstructured":"Cand\u00e9s, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717\u2013772 (2009)"},{"issue":"5\u20136","key":"9933_CR7","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1007\/s00041-008-9045-x","volume":"14","author":"EJ Cand\u00e9s","year":"2008","unstructured":"Cand\u00e9s, E.J., Wakin, M.B., Boyd, S.P.: Enhancing sparsity by reweighted $$\\ell _1$$ \u2113 1 minimization. J. Fourier Anal. Appl. 14(5\u20136), 877\u2013905 (2008)","journal-title":"J. Fourier Anal. Appl."},{"key":"9933_CR8","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1109\/LSP.2007.898300","volume":"14","author":"R Chartrand","year":"2007","unstructured":"Chartrand, R.: Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process. Lett. 14, 707\u2013710 (2007)","journal-title":"IEEE Signal Process. Lett."},{"key":"9933_CR9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1088\/0266-5611\/24\/3\/035020","volume":"24","author":"R Chartrand","year":"2008","unstructured":"Chartrand, R., Staneva, V.: Restricted isometry properties and nonconvex compressive sensing. Inverse Probl. 24, 1\u201314 (2008)","journal-title":"Inverse Probl."},{"key":"9933_CR10","doi-asserted-by":"crossref","unstructured":"Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: 33rd International Conference on Acoustics, Speech, and Signal Processing (ICASSP) (2008)","DOI":"10.1109\/ICASSP.2008.4518498"},{"issue":"1","key":"9933_CR11","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s10107-012-0613-0","volume":"143","author":"X Chen","year":"2014","unstructured":"Chen, X., Ge, D., Wang, Z., Ye, Y.: Complexity of unconstrained $$L_2$$ L 2 - $$L_p$$ L p minimization. Math. Program. 143(1), 371\u2013383 (2014)","journal-title":"Math. Program."},{"issue":"3","key":"9933_CR12","doi-asserted-by":"crossref","first-page":"1528","DOI":"10.1137\/120871390","volume":"23","author":"X Chen","year":"2013","unstructured":"Chen, X., Niu, L., Yuan, Y.: Optimality conditions and a smoothing trust region Newton method for non-Lipschitz optimization. SIAM J. Optim. 23(3), 1528\u20131552 (2013)","journal-title":"SIAM J. Optim."},{"key":"9933_CR13","doi-asserted-by":"crossref","first-page":"2832","DOI":"10.1137\/090761471","volume":"32","author":"X Chen","year":"2010","unstructured":"Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of $$l_2$$ l 2 - $$l_p$$ l p minimization. SIAM J. Sci. Comput. 32, 2832\u20132852 (2010)","journal-title":"SIAM J. Sci. Comput."},{"key":"9933_CR14","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s10589-013-9553-8","volume":"59","author":"X Chen","year":"2014","unstructured":"Chen, X., Zhou, W.: Convergence of reweighted $$l_1$$ l 1 minimization algorithm for $$l_2$$ l 2 - $$l_p$$ l p minimization. Comput. Optim. Appl. 59, 47\u201361 (2014)","journal-title":"Comput. Optim. Appl."},{"key":"9933_CR15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/cpa.20303","volume":"63","author":"I Daubechies","year":"2010","unstructured":"Daubechies, I., DeVore, R., Fornasier, M., Gunturk, C.S.: Iteratively reweighted least squares minimization for sparse recovery. Commun. Pure. Appl. Math. 63, 1\u201338 (2010)","journal-title":"Commun. Pure. Appl. Math."},{"key":"9933_CR16","doi-asserted-by":"crossref","first-page":"4734","DOI":"10.1109\/ACC.2001.945730","volume":"6","author":"M Fazel","year":"2001","unstructured":"Fazel, M., Hindi, H., Boyd, S.P.: A rank minimization heuristic with application to minimum order system approximation. P. Am. Contr. Conf. 6, 4734\u20134739 (2001)","journal-title":"P. Am. Contr. Conf."},{"issue":"456","key":"9933_CR17","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1198\/016214501753382273","volume":"96","author":"J Fan","year":"2001","unstructured":"Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348\u20131360 (2001)","journal-title":"J. Am. Stat. Assoc."},{"key":"9933_CR18","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.acha.2008.09.001","volume":"26","author":"S Foucart","year":"2009","unstructured":"Foucart, S., Lai, M.: Sparsest solutions of underdetermined linear systems via $$l_q$$ l q -minimization for $$0 < q \\le 1$$ 0 < q \u2264 1 . Appl. Comput. Harmon. Anal. 26, 395\u2013407 (2009)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9933_CR19","doi-asserted-by":"crossref","first-page":"1614","DOI":"10.1137\/100811404","volume":"21","author":"M Fornasier","year":"2011","unstructured":"Fornasier, M., Rauhut, H., Ward, R.: Low-rank matrix recovery via iteratively reweighted least squares minimization. SIAM J. Optim. 21, 1614\u20131640 (2011)","journal-title":"SIAM J. Optim."},{"key":"9933_CR20","first-page":"1721","volume":"21","author":"D Ge","year":"2011","unstructured":"Ge, D., Jiang, X., Ye, Y.: A note on the complexity of $$L_p$$ L p minimization. Math. Program. 21, 1721\u20131739 (2011)","journal-title":"Math. Program."},{"key":"9933_CR21","unstructured":"Gong, P., Zhang, C., Lu, Z., Huang, J., Ye, J.: A general iterative shrinkage and thresholding algorithm for non-convex regularized optimization problems. In: The 30th International Conference on Machine Learning (ICML) (2013)"},{"key":"9933_CR22","doi-asserted-by":"crossref","unstructured":"Ji, S., Sze, K.-F., Zhou, Z., So, A., Ye, Y.: Beyond convex relaxation: a polynomialtime nonconvex optimization approach to network localization. In: INFOCOM (2013)","DOI":"10.1109\/INFCOM.2013.6567056"},{"issue":"3","key":"9933_CR23","doi-asserted-by":"crossref","first-page":"1340010","DOI":"10.1142\/S0217595913400101","volume":"30","author":"L Kong","year":"2013","unstructured":"Kong, L., Xiu, N.: Exact low-rank matrix recovery via nonconvex Schatten $$p$$ p -minimization. Asia Pac. J. Oper. Res. 30(3), 1340010 (2013)","journal-title":"Asia Pac. J. Oper. Res."},{"key":"9933_CR24","unstructured":"Lai, M., Li, S., Liu, L.Y., Wang, H.: Two results on the schatten $$p$$ p -quasi-norm minimization for low-rank matrix recovery. Preprint (2012)"},{"key":"9933_CR25","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1137\/090775397","volume":"21","author":"M Lai","year":"2010","unstructured":"Lai, M., Wang, J.: An unconstrained $$l_q$$ l q minimization with $$0 < q \\le 1$$ 0 < q \u2264 1 for sparse solution of underdetermined linear systems. SIAM J. Optim. 21, 82\u2013101 (2010)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"9933_CR26","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1137\/110840364","volume":"5","author":"M Lai","year":"2013","unstructured":"Lai, M., Xu, Y., Yin, W.: Improved iteratively reweighted least squares for unconstrained smoothed $$l_q$$ l q minimization. SIAM J. Numer. Anal. 5(2), 927\u2013957 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"9933_CR27","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s11228-004-7197-7","volume":"13","author":"AS Lewis","year":"2005","unstructured":"Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part I: theory. Set Valued Anal. 13, 213\u2013241 (2005)","journal-title":"Set Valued Anal."},{"key":"9933_CR28","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s11228-004-7198-6","volume":"13","author":"AS Lewis","year":"2005","unstructured":"Lewis, A.S., Sendov, H.S.: Nonsmooth analysis of singular values. Part II: applications. Set Valued Anal. 13, 243\u2013264 (2005)","journal-title":"Set Valued Anal."},{"issue":"1\u20132","key":"9933_CR29","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s10107-013-0722-4","volume":"147","author":"Z Lu","year":"2014","unstructured":"Lu, Z.: Iterative reweighted minimization methods for $$l_p$$ l p regularized unconstrained nonlinear programming. Math. Program. 147(1\u20132), 277\u2013307 (2014)","journal-title":"Math. Program."},{"issue":"3","key":"9933_CR30","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1080\/10556788.2014.936438","volume":"30","author":"Z Lu","year":"2015","unstructured":"Lu, Z., Zhang, Y., Li, X.: Penalty decomposition methods for rank minimization. Optim. Method Softw. 30(3), 531\u2013558 (2015)","journal-title":"Optim. Method Softw."},{"issue":"1","key":"9933_CR31","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s10107-009-0306-5","volume":"128","author":"S Ma","year":"2011","unstructured":"Ma, S., Goldfarb, D., Chen, L.: Fixed point and Bregman iterative methods for matrix rank minimization. Math. Program. 128(1), 321\u2013353 (2011)","journal-title":"Math. Program."},{"key":"9933_CR32","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.sigpro.2015.02.025","volume":"114","author":"M Malek-Mohammadi","year":"2015","unstructured":"Malek-Mohammadi, M., Babaie-Zadeh, M., Skoglund, M.: Performance guarantees for Schatten- $$p$$ p quasi-norm minimization in recovery of low-rank matrices. Sig. Process. 114, 225\u2013230 (2015)","journal-title":"Sig. Process."},{"key":"9933_CR33","doi-asserted-by":"crossref","unstructured":"Mohan, K., Fazel, M.: Iterative reweighted least squares for matrix rank minimization. In: 48th Annual Allerton Conference on Communication, Control, and Computing, pp. 653\u2013661 (2010)","DOI":"10.1109\/ALLERTON.2010.5706969"},{"key":"9933_CR34","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1109\/34.608289","volume":"19","author":"T Mrita","year":"1997","unstructured":"Mrita, T., Kanade, T.: A sequential factorization method for recovering shape and motion from image streams. IEEE T. Pattern Anal. 19, 858\u2013867 (1997)","journal-title":"IEEE T. Pattern Anal."},{"key":"9933_CR35","doi-asserted-by":"crossref","first-page":"3485","DOI":"10.1109\/TSP.2010.2046900","volume":"58","author":"N Mourad","year":"2010","unstructured":"Mourad, N., Reilly, J.P.: Minimizing nonconvex functions for sparse vector reconstruction. IEEE Trans. Signal Process. 58, 3485\u20133496 (2010)","journal-title":"IEEE Trans. Signal Process."},{"key":"9933_CR36","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1109\/78.738251","volume":"47","author":"BD Rao","year":"1999","unstructured":"Rao, B.D., Kreutz-Delgado, K.: An affine scaling methodology for best basis selection. IEEE Trans. Signal Process. 47, 187\u2013200 (1999)","journal-title":"IEEE Trans. Signal Process."},{"issue":"3","key":"9933_CR37","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1137\/070697835","volume":"52","author":"B Recht","year":"2010","unstructured":"Recht, B., Fazel, M., Parrilo, P.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471\u2013501 (2010)","journal-title":"SIAM Rev."},{"key":"9933_CR38","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.acha.2011.07.001","volume":"32","author":"Q Sun","year":"2012","unstructured":"Sun, Q.: Recovery of sparsest signals via $$l_q$$ l q minimization. Appl. Comput. Harmon. Anal. 32, 329\u2013341 (2012)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9933_CR39","first-page":"615","volume":"6","author":"K Toh","year":"2010","unstructured":"Toh, K., Yun, S.: An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. Pac. J. Optim. 6, 615\u2013640 (2010)","journal-title":"Pac. J. Optim."},{"key":"9933_CR40","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF00129684","volume":"9","author":"C Tpmasi","year":"1992","unstructured":"Tpmasi, C., Kanade, T.: Shape and motion from image streams under orthography: a factorization method. Int. J. Comput. Vis. 9, 137\u2013154 (1992)","journal-title":"Int. J. Comput. Vis."},{"key":"9933_CR41","doi-asserted-by":"crossref","first-page":"A488","DOI":"10.1137\/130934271","volume":"37","author":"Z Wang","year":"2015","unstructured":"Wang, Z., Lai, M.-J., Lu, Z., Fan, W., Davulcu, H., Ye, J.: Orthogonal rank-one matrix pursuit for low rank matrix completion. SIAM J. Sci. Comput. 37, A488\u2013A514 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"9933_CR42","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/s12532-012-0044-1","volume":"4","author":"Z Wen","year":"2012","unstructured":"Wen, Z., Yin, W., Zhang, Y.: Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm. Math. Prog. Comput. 4, 333\u2013361 (2012)","journal-title":"Math. Prog. Comput."},{"key":"9933_CR43","doi-asserted-by":"crossref","first-page":"2479","DOI":"10.1109\/TSP.2009.2016892","volume":"57","author":"SJ Wright","year":"2009","unstructured":"Wright, S.J., Nowak, R., Figueiredo, M.: Sparse reconstruction by separable approximation. IEEE T. Image Process. 57, 2479\u20132493 (2009)","journal-title":"IEEE T. Image Process."},{"key":"9933_CR44","first-page":"1","volume":"52","author":"Z Xu","year":"2009","unstructured":"Xu, Z., Zhang, H., Wang, Y., Chang, X.: $$L_{1\/2}$$ L 1 \/ 2 regularizer. Sci. China 52, 1\u20139 (2009)","journal-title":"Sci. China"},{"key":"9933_CR45","unstructured":"Yue, M., So, A.M.: A perturbation inequality for the schatten- $$p$$ p quasi-norm and its applications to low-rank matrix recovery. Appl. Comput. Harmon. Anal. 40, 396\u2013416 (2016)"},{"issue":"2","key":"9933_CR46","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1214\/09-AOS729","volume":"38","author":"CH Zhang","year":"2010","unstructured":"Zhang, C.H.: Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 38(2), 894\u2013942 (2010)","journal-title":"Ann. Stat."},{"key":"9933_CR47","first-page":"1081","volume":"11","author":"T Zhang","year":"2010","unstructured":"Zhang, T.: Analysis of multi-stage convex relaxation for sparse regularization. J. Mach. Learn. Res. 11, 1081\u20131107 (2010)","journal-title":"J. Mach. Learn. Res."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-017-9933-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-017-9933-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-017-9933-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,2]],"date-time":"2019-10-02T01:18:39Z","timestamp":1569979119000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-017-9933-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,9]]},"references-count":47,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,12]]}},"alternative-id":["9933"],"URL":"https:\/\/doi.org\/10.1007\/s10589-017-9933-6","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"type":"print","value":"0926-6003"},{"type":"electronic","value":"1573-2894"}],"subject":[],"published":{"date-parts":[[2017,8,9]]}}}