{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T09:26:45Z","timestamp":1787390805922,"version":"build-2736575974"},"reference-count":44,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee","doi-asserted-by":"publisher","award":["PolyU153085\/16p"],"award-info":[{"award-number":["PolyU153085\/16p"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee","doi-asserted-by":"publisher","award":["PolyU153000\/15p"],"award-info":[{"award-number":["PolyU153000\/15p"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In this paper we consider a general matrix factorization model which covers a large class of existing models with many applications in areas such as machine learning and imaging sciences. To solve this possibly nonconvex, nonsmooth, and non-Lipschitz problem, we develop a nonmonotone alternating updating method based on a potential function. Our method essentially updates two blocks of variables in turn by inexactly minimizing this potential function, and updates another auxiliary block of variables using an explicit formula. The special structure of our potential function allows us to take advantage of efficient computational strategies for nonnegative matrix factorization to perform the alternating minimization over the two blocks of variables. A suitable line search criterion is also incorporated to improve the numerical performance. Under some mild conditions, we show that the line search criterion is well defined, and establish that the sequence generated is bounded and any cluster point of the sequence is a stationary point. Finally, we conduct some numerical experiments using real datasets to compare our method with some existing efficient methods for nonnegative matrix factorization and matrix completion. The numerical results show that our method can outperform these methods for these specific applications.<\/jats:p>","DOI":"10.1137\/17m1130113","type":"journal-article","created":{"date-parts":[[2018,12,13]],"date-time":"2018-12-13T11:48:37Z","timestamp":1544701717000},"page":"3402-3430","source":"Crossref","is-referenced-by-count":7,"title":["A Nonmonotone Alternating Updating Method for a Class of Matrix Factorization Problems"],"prefix":"10.1137","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9979-8371","authenticated-orcid":true,"given":"Lei","family":"Yang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ting Kei","family":"Pong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaojun","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,12,13]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-011-0484-9"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2006.11.006"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1109\/TASE.2006.877401"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0701-9"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-009-9045-5"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2010.2044061"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/15M1028054"},{"key":"atypb8","first-page":"169","author":"Cichocki A.","year":"2007","journal-title":"New York"},{"key":"atypb9","unstructured":"N. Gillis,\n                      Nonnegative Matrix Factorization: Complexity, Algorithms and Applications\n                      , Ph.D. thesis, Universit\u00e9 catholique de Louvain, 2011."},{"key":"atypb10","first-page":"257","author":"Gillis N.","year":"2014","journal-title":"Chapman and Hall\/CRC"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1162\/NECO_a_00256"},{"key":"atypb12","first-page":"37","volume":"28","author":"Gong P.","year":"2013","journal-title":"Proc. Mach. Learn. Res."},{"key":"atypb13","unstructured":"M. Hong,\n                      Decomposing Linearly Constrained Nonconvex Problems by a Proximal Primal Dual Approach: Algorithms, Convergence and Applications\n                      , preprint,https:\/\/arxiv.org\/abs\/1604.00543, 2016."},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"B. Jiang, T. Lin, S. Ma, and S. Zhang,\n                      Structured nonconvex and nonsmooth optimization: Algorithms and iteration complexity analysis\n                      , Comput. Optim. Appl. (2018),https:\/\/doi.org\/10.1007\/s10589-018-0034-y.","DOI":"10.1007\/s10589-018-0034-y"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2010.2046205"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/110840364"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1038\/44565"},{"key":"atypb18","first-page":"556","author":"Lee D.","year":"2001","journal-title":"MA"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2005.92"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1117\/1.3184771"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.patcog.2011.05.015"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/090755436"},{"key":"atypb23","first-page":"3441","volume":"13","author":"Mohan K.","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"atypb24","unstructured":"P. Ochs and T. Pock,\n                      Adaptive FISTA\n                      , preprint,https:\/\/arxiv.org\/abs\/1711.04343, 2017."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1002\/env.3170050203"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/070697835"},{"key":"atypb27","first-page":"713","author":"Rennie J.","year":"2005","journal-title":"New York"},{"key":"atypb28","doi-asserted-by":"crossref","unstructured":"R. Rockafellar and R.B. Wets,\n                      Variational Analysis\n                      , Springer, New York, 1998.","DOI":"10.1007\/978-3-642-02431-3"},{"key":"atypb29","first-page":"138","author":"Samaria F.","year":"1994","journal-title":"NJ"},{"key":"atypb30","doi-asserted-by":"crossref","unstructured":"F. Shang, Y. Liu, and J. Cheng,\n                      Scalable algorithms for tractable Schatten quasi-norm minimization\n                      , in Proceedings of the 30th AAAI Conference on Artificial Intelligence, Association for the Advancement of Artificial Intelligence, Palo Alto, CA, 2016, pp. 2016-2022.","DOI":"10.1609\/aaai.v30i1.10266"},{"key":"atypb31","unstructured":"F. Shang, Y. Liu, and J. Cheng,\n                      Tractable and scalable Schatten quasi-norm approximations for rank minimization\n                      , in Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, Association for the Advancement of Artificial Intelligence, Palo Alto, CA, 2016, pp. 620-629."},{"key":"atypb32","unstructured":"F. Shang, Y. Liu, and J. Cheng,\n                      Unified Scalable Equivalent Formulations for Schatten Quasi-norms\n                      , preprint,https:\/\/arxiv.org\/abs\/1606.00668, 2016."},{"key":"atypb33","unstructured":"N. Srebro,\n                      Learning with Matrix Factorizations\n                      , Ph.D. thesis, Massachusetts Institute of Technology, 2004."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2016.2598574"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1561\/2200000055"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1137\/070709967"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1109\/TKDE.2012.51"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1007\/s12532-012-0044-1"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2009.2016892"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1137\/120887795"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1007\/s11464-012-0194-5"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2013.2250577"},{"key":"atypb43","unstructured":"Y. Zhang,\n                      An Alternating Direction Algorithm for Nonnegative Matrix Factorization\n                      , preprint,https:\/\/scholarship.rice.edu\/bitstream\/handle\/1911\/102146\/TR10-03.pdf?sequence=1, 2010."},{"key":"atypb44","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1145\/2674559","volume":"47","author":"Zhou X.","year":"2015","journal-title":"ACM Computing Surveys"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1130113","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:53:03Z","timestamp":1787338383000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1130113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":44,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1130113"],"URL":"https:\/\/doi.org\/10.1137\/17m1130113","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}