{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T20:57:45Z","timestamp":1761944265699,"version":"build-2065373602"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T00:00:00Z","timestamp":1753315200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T00:00:00Z","timestamp":1753315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100003453","name":"Natural Science Foundation of Guangdong Province","doi-asserted-by":"publisher","award":["2023A1515111167"],"award-info":[{"award-number":["2023A1515111167"]}],"id":[{"id":"10.13039\/501100003453","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12371299"],"award-info":[{"award-number":["12371299"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2025,11]]},"DOI":"10.1007\/s10589-025-00708-6","type":"journal-article","created":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T13:25:44Z","timestamp":1753363544000},"page":"617-653","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Convergence of the majorized PAM method with subspace correction for low-rank composite factorization model"],"prefix":"10.1007","volume":"92","author":[{"given":"Ting","family":"Tao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yitian","family":"Qian","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shaohua","family":"Pan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,24]]},"reference":[{"key":"708_CR1","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.jvcir.2012.10.006","volume":"24","author":"W Cao","year":"2013","unstructured":"Cao, W., Sun, J., Xu, Z.: Fast image deconvolution using closed-form thresholding formulas of $$L_q (q=\\frac{1}{2},\\frac{2}{3})$$ regularization. J. Vis. Commun. Image Represent. 24, 31\u201341 (2013)","journal-title":"J. Vis. Commun. Image Represent."},{"key":"708_CR2","doi-asserted-by":"publisher","first-page":"1013","DOI":"10.1109\/TNNLS.2012.2197412","volume":"23","author":"Z Xu","year":"2012","unstructured":"Xu, Z., Chang, X., Xu, F., Zhang, H.: $$\\ell _{1\/2}$$ regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learn. Syst. 23, 1013\u20131027 (2012)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"708_CR3","doi-asserted-by":"publisher","first-page":"959","DOI":"10.1137\/20M136205X","volume":"32","author":"T Tao","year":"2022","unstructured":"Tao, T., Qian, Y.T., Pan, S.H.: Column $$\\ell _{2,0}$$-norm regularized factorization model of low-rank matrix recovery and its computation. SIAM J. Optim. 32, 959\u2013988 (2022)","journal-title":"SIAM J. Optim."},{"key":"708_CR4","doi-asserted-by":"publisher","first-page":"3098","DOI":"10.1137\/19M1290000","volume":"30","author":"YX Chen","year":"2020","unstructured":"Chen, Y.X., Chi, Y.J., Fan, J.Q., Ma, C., Yan, Y.L.: Noisy matrix completion: understanding statistical guarantees for convex relaxation via nonconvex optimization. SIAM J. Optim. 30, 3098\u20133121 (2020)","journal-title":"SIAM J. Optim."},{"key":"708_CR5","first-page":"3367","volume":"16","author":"T Hastie","year":"2015","unstructured":"Hastie, T., Mazumder, R., Lee, J.D., Zadeh, R.: Matrix completion and low-rank SVD via fast alternating least squares. J. Mach. Learn. Res. 16, 3367\u20133402 (2015)","journal-title":"J. Mach. Learn. Res."},{"key":"708_CR6","first-page":"3873","volume":"29","author":"S Bhojanapalli","year":"2016","unstructured":"Bhojanapalli, S., Neyshabur, B., Srebro, N.: Global optimality of local search for low rank matrix recovery. Adv. Neural Inf. Process. Syst. 29, 3873\u20133881 (2016)","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"708_CR7","first-page":"3619","volume":"9","author":"T Cai","year":"2013","unstructured":"Cai, T., Zhou, W.X.: A max-norm constrained minimization approach to 1-bit matrix completion. J. Mach. Learn. Res. 9, 3619\u20133647 (2013)","journal-title":"J. Mach. Learn. Res."},{"doi-asserted-by":"crossref","unstructured":"Jain, P., Netrapalli, P., Sanghavi, S.: Low-rank matrix completion using alternating minimization. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC), 665-674 (2013)","key":"708_CR8","DOI":"10.1145\/2488608.2488693"},{"issue":"11","key":"708_CR9","doi-asserted-by":"publisher","first-page":"6535","DOI":"10.1109\/TIT.2016.2598574","volume":"26","author":"RY Sun","year":"2016","unstructured":"Sun, R.Y., Luo, Z.Q.: Guaranteed matrix completion via non-convex factorization. IEEE Trans. Inf. Theory 26(11), 6535\u20136579 (2016)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"708_CR10","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1287\/moor.1100.0449","volume":"35","author":"H Attouch","year":"2010","unstructured":"Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kerdyka-\u0141ojasiewicz inequality. Math. Oper. Res. 35, 438\u2013457 (2010)","journal-title":"Math. Oper. Res."},{"key":"708_CR11","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1007\/s10107-013-0701-9","volume":"146","author":"J Bolte","year":"2014","unstructured":"Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146, 459\u2013494 (2014)","journal-title":"Math. Program."},{"key":"708_CR12","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1007\/s10898-016-0405-9","volume":"66","author":"E Chouzenoux","year":"2016","unstructured":"Chouzenoux, E., Pesque, J.C., Repetti, A.: A block coordinate variable metric forward-backward algorithm. J. Global Optim. 66, 457\u2013485 (2016)","journal-title":"J. Global Optim."},{"key":"708_CR13","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1137\/17M1124085","volume":"29","author":"P Ochs","year":"2019","unstructured":"Ochs, P.: Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano. SIAM J. Optim. 29, 541\u2013570 (2019)","journal-title":"SIAM J. Optim."},{"key":"708_CR14","first-page":"1756","volume":"9","author":"T Pock","year":"2016","unstructured":"Pock, T., Sabach, S.: Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems. SIAM Rev. 9, 1756\u20131787 (2016)","journal-title":"SIAM Rev."},{"key":"708_CR15","doi-asserted-by":"publisher","first-page":"700","DOI":"10.1007\/s10915-017-0376-0","volume":"72","author":"YY Xu","year":"2017","unstructured":"Xu, Y.Y., Yin, W.T.: A globally convergent algorithm for nonconvex optimization based on block coordinate update. J. Sci. Comput. 72, 700\u2013734 (2017)","journal-title":"J. Sci. Comput."},{"key":"708_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational analysis","author":"RT Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer, New York (1998)"},{"key":"708_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3390\/math8081325","volume":"8","author":"FH Shang","year":"2020","unstructured":"Shang, F.H., Liu, Y.Y., Shang, F.J.: A unified scalable equivalent formulation for Schatten quasi-norms. Mathematics 8, 1\u201319 (2020)","journal-title":"Mathematics"},{"key":"708_CR18","first-page":"1","volume":"18","author":"SJ Bi","year":"2022","unstructured":"Bi, S.J., Tao, T., Pan, S.H.: KL property of exponent $$1\/2$$ of $$\\ell _{2,0}$$-norm and DC regularized factorizations for low-rank matrix recovery. Pac. J. Optim. 18, 1\u201326 (2022)","journal-title":"Pac. J. Optim."},{"key":"708_CR19","first-page":"1","volume":"17","author":"N Srebro","year":"2005","unstructured":"Srebro, N., Rennie, J.D.M., Jaakkola, T.: Maximum-margin matrix factorization. Adv. Neural Inf. Process. Syst. 17, 1 (2005)","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"708_CR20","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1007\/s10107-012-0619-7","volume":"144","author":"C Ding","year":"2014","unstructured":"Ding, C., Sun, D.F., Toh, K.C.: An introduction to a class of matrix cone programming. Math. Program. 144, 141\u2013179 (2014)","journal-title":"Math. Program."},{"issue":"2","key":"708_CR21","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1023\/A:1022303426500","volume":"40","author":"FM Dopico","year":"2000","unstructured":"Dopico, F.M.: A note on sin theorems for singular subspace variations. BIT Numer. Math. 40(2), 395\u2013403 (2000)","journal-title":"BIT Numer. Math."},{"key":"708_CR22","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1007\/s10107-002-0306-1","volume":"95","author":"X Chen","year":"2003","unstructured":"Chen, X., Tseng, P.: Non-Interior continuation methods for solving semidefinite complementarity problems. Math. Program. 95, 431\u2013474 (2003)","journal-title":"Math. Program."},{"key":"708_CR23","doi-asserted-by":"publisher","first-page":"991","DOI":"10.1007\/s10898-021-01077-0","volume":"81","author":"T Tao","year":"2021","unstructured":"Tao, T., Pan, S.H., Bi, S.J.: Error bound of critical points and KL property of exponent $$1\/2$$ for squared F-norm regularized factorization. J. Global Optim. 81, 991\u20131017 (2021)","journal-title":"J. Global Optim."},{"key":"708_CR24","volume-title":"Maximum-margin matrix factorization","author":"GW Stewart","year":"1990","unstructured":"Stewart, G.W., Sun, J.G.: Maximum-margin matrix factorization. Academic Press, Boston (1990)"},{"key":"708_CR25","doi-asserted-by":"publisher","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."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00708-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-025-00708-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00708-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T20:55:16Z","timestamp":1761944116000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-025-00708-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,24]]},"references-count":25,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["708"],"URL":"https:\/\/doi.org\/10.1007\/s10589-025-00708-6","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"type":"print","value":"0926-6003"},{"type":"electronic","value":"1573-2894"}],"subject":[],"published":{"date-parts":[[2025,7,24]]},"assertion":[{"value":"9 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 June 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 July 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}