{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T16:57:14Z","timestamp":1773075434564,"version":"3.50.1"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2012,9,26]],"date-time":"2012-09-26T00:00:00Z","timestamp":1348617600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2013,5]]},"DOI":"10.1007\/s10589-012-9507-6","type":"journal-article","created":{"date-parts":[[2012,9,26]],"date-time":"2012-09-26T01:26:11Z","timestamp":1348622771000},"page":"173-196","source":"Crossref","is-referenced-by-count":16,"title":["Modified subspace Barzilai-Borwein gradient method for non-negative matrix factorization"],"prefix":"10.1007","volume":"55","author":[{"given":"Hongwei","family":"Liu","sequence":"first","affiliation":[]},{"given":"Xiangli","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2012,9,26]]},"reference":[{"key":"9507_CR1","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1038\/44565","volume":"401","author":"D.D. Lee","year":"1999","unstructured":"Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401, 788\u2013791 (1999)","journal-title":"Nature"},{"key":"9507_CR2","volume-title":"Proceedings of the Fourth SIAM International Conference on Data Mining","author":"V.P. Pauca","year":"2004","unstructured":"Pauca, V.P., Shahnaz, F., Berry, M.W., Plemmons, R.J.: Text mining using non-negative matrix factorization. In: Proceedings of the Fourth SIAM International Conference on Data Mining (2004)"},{"key":"9507_CR3","doi-asserted-by":"crossref","first-page":"1706","DOI":"10.1101\/gr.903503","volume":"13","author":"P.M. Kim","year":"2003","unstructured":"Kim, P.M., Tidor, B.: Subsystem identification through dimensionality reduction of large-scale gene expression data. Genome Res. 13, 1706\u20131718 (2003)","journal-title":"Genome Res."},{"key":"9507_CR4","first-page":"4164","volume-title":"Proceedings of the National Academy of Sciences of the United States of America","author":"J.-P. Brunet","year":"2004","unstructured":"Brunet, J.-P., Tamayo, P., Golub, T.R., Mesirov, J.P.: Metagenes and molecular pattern discovery using matrix factorization. In: Proceedings of the National Academy of Sciences of the United States of America, vol. 101, pp. 4164\u20134169 (2004)"},{"key":"9507_CR5","doi-asserted-by":"crossref","first-page":"3970","DOI":"10.1093\/bioinformatics\/bti653","volume":"21","author":"Y. Gao","year":"2005","unstructured":"Gao, Y., Church, G.: Improving molecular cancer class discovery through sparse non-negative matrix factorization. Bioinformatics 21, 3970\u20133975 (2005)","journal-title":"Bioinformatics"},{"key":"9507_CR6","doi-asserted-by":"crossref","first-page":"1495","DOI":"10.1093\/bioinformatics\/btm134","volume":"23","author":"H. Kim","year":"2007","unstructured":"Kim, H., Park, H.: Sparse non-negative matrix factorizations via alternating non-negativity-constrained least squares for microarray data analysis. Bioinformatics 23, 1495\u20131502 (2007)","journal-title":"Bioinformatics"},{"key":"9507_CR7","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1364\/JOSA.62.000055","volume":"62","author":"W.H. Richardson","year":"1972","unstructured":"Richardson, W.H.: Bayesian-based iterative method of image restoration. J. Opt. Soc. Am. 62, 55\u201359 (1972)","journal-title":"J. Opt. Soc. Am."},{"key":"9507_CR8","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.1016\/j.patrec.2004.11.026","volume":"26","author":"Y.-C. Cho","year":"2005","unstructured":"Cho, Y.-C., Choi, S.: Nonnegative features of spectro-temporal sounds for classification. Pattern Recognit. Lett. 26, 1327\u20131336 (2005)","journal-title":"Pattern Recognit. Lett."},{"key":"9507_CR9","volume-title":"Computational Systems Bioinformatics Conference","author":"N. Rao","year":"2004","unstructured":"Rao, N., Shepherd, S.J.: Extracting characteristic patterns from genome\u2014wide expression data by non-negative matrix factorization. In: Computational Systems Bioinformatics Conference (2004)"},{"key":"9507_CR10","doi-asserted-by":"crossref","first-page":"2447","DOI":"10.1016\/S0167-8655(03)00089-8","volume":"24","author":"D. Guillamet","year":"2003","unstructured":"Guillamet, D., Vitria, J., Schiele, B.: Introducing a weighted non-negative matrix factorization for image classification. Pattern Recognit. Lett. 24, 2447\u20132454 (2003)","journal-title":"Pattern Recognit. Lett."},{"key":"9507_CR11","first-page":"116","volume-title":"16th International Conference on Pattern Recognition","author":"D. Guillamet","year":"2002","unstructured":"Guillamet, D., Schiele, B., Vitri\u00e0, J.: Analyzing non-negative matrix factorization for image classification. In: 16th International Conference on Pattern Recognition, vol. 2, pp. 116\u2013119 (2002)"},{"key":"9507_CR12","volume-title":"Procedure of the Fifth Catalan Conference for Artificial Intelligence","author":"D. Guillamet","year":"2002","unstructured":"Guillamet, D., Vitria, J.: Classifying faces with nonnegative matrix factorization. In: Procedure of the Fifth Catalan Conference for Artificial Intelligence (2002)"},{"key":"9507_CR13","volume-title":"ACCV","author":"J.-H. Ahn","year":"2004","unstructured":"Ahn, J.-H., Kim, S.-K., Oh, J.-H., Choi, S.: Multiple nonnegativematrix factorization of dynamic PET images. In: ACCV (2004)"},{"key":"9507_CR14","first-page":"556","volume-title":"3rd International Conference on Independent Component Analysis and Blind Signal Separation","author":"J.S. Lee","year":"2002","unstructured":"Lee, J.S., Lee, D.D., Choi, S., Lee, D.S.: Application of nonnegative matrix factorization to dynamic positron emission tomography. In: 3rd International Conference on Independent Component Analysis and Blind Signal Separation, pp. 556\u2013562 (2002)"},{"key":"9507_CR15","first-page":"253","volume-title":"IEEE Workshop on Machine Learning for Signal Processing","author":"H. Li","year":"2005","unstructured":"Li, H., Adali, T., Wang, W., Emge, D.: Non-negative matrix factorization with orthogonality constraints for chemical agent detection in Raman spectra. In: IEEE Workshop on Machine Learning for Signal Processing, pp. 253\u2013258 (2005)"},{"key":"9507_CR16","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1109\/TPAMI.2006.60","volume":"28","author":"K. Kochi","year":"2006","unstructured":"Kochi, K., Lehmann, D., Pascual\u00a0Marqui, R.D.: Nonsmooth nonnegative matrix factorization (nsNMF). IEEE Trans. Pattern Anal. Mach. Intell. 28, 403\u2013415 (2006)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"9507_CR17","first-page":"62","volume":"2006","author":"O. Okun","year":"2009","unstructured":"Okun, O., Priisalu, H.: Fast nonnegative matrix factorization and its application for protein fold recognition. EURASIP J. Appl. Signal Process. 2006, 62 (2009)","journal-title":"EURASIP J. Appl. Signal Process."},{"key":"9507_CR18","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1142\/S0218001405004198","volume":"19","author":"Y. Wang","year":"2005","unstructured":"Wang, Y., Jia, Y., Hu, C., Matthew, T.: Non-negative matrix factorization framework for face recognition. Int. J. Pattern Recognit. Artif. Intell. 19, 495\u2013511 (2005)","journal-title":"Int. J. Pattern Recognit. Artif. Intell."},{"key":"9507_CR19","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1016\/j.patrec.2004.02.002","volume":"25","author":"W. Liu","year":"2004","unstructured":"Liu, W., Zheng, N.: Non-negative matrix factorization based methods for object recognition. Pattern Recognit. Lett. 25, 893\u2013897 (2004)","journal-title":"Pattern Recognit. Lett."},{"key":"9507_CR20","first-page":"793","volume":"7","author":"M.W. Spratling","year":"2006","unstructured":"Spratling, M.W.: Learning image components for object recognition. J. Mach. Learn. Res. 7, 793\u2013815 (2006)","journal-title":"J. Mach. Learn. Res."},{"key":"9507_CR21","first-page":"556","volume-title":"Proceedings of Neural Information Processing Systems","author":"D.D. Lee","year":"2001","unstructured":"Lee, D.D., Seung, H.S.: Algorithms for non-negative matrix factorization. In: Proceedings of Neural Information Processing Systems, vol. 13, pp. 556\u2013562 (2001)"},{"key":"9507_CR22","unstructured":"Chu, M., Diele, F., Plemmons, R., Ragni, S.: Optimality, computation, and interpretations of nonnegative matrix factorizations. SIAM J. Matrix Anal. 4\u20138030 (2004)"},{"key":"9507_CR23","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1002\/env.3170050203","volume":"5","author":"P. Paatero","year":"1994","unstructured":"Paatero, P., Tapper, U.: Positive matrix factorization: a non-negative factor model with optimal utilization of error estimates of data values. Environmetrics 5, 111\u2013126 (1994)","journal-title":"Environmetrics"},{"key":"9507_CR24","doi-asserted-by":"crossref","first-page":"2756","DOI":"10.1162\/neco.2007.19.10.2756","volume":"19","author":"C.-J. Lin","year":"2007","unstructured":"Lin, C.-J.: Projected gradient methods for nonnegative matrix factorization. Neural Comput. 19, 2756\u20132779 (2007)","journal-title":"Neural Comput."},{"key":"9507_CR25","volume-title":"Proceedings of AMOS Technical Conference","author":"J. Piper","year":"2004","unstructured":"Piper, J., Pauca, P., Plemmons, R., Giffin, M.: Object characterization from spectral data using nonnegative factorization and information theory. In: Proceedings of AMOS Technical Conference (2004)"},{"key":"9507_CR26","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1109\/TNN.2007.895831","volume":"18","author":"C.-J. Lin","year":"2007","unstructured":"Lin, C.-J.: On the convergence of multiplicative update algorithms for non-negative matrix factorization. IEEE Trans. Neural Netw. 18, 1589\u20131596 (2007)","journal-title":"IEEE Trans. Neural Netw."},{"key":"9507_CR27","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.ipm.2004.11.005","volume":"42","author":"F. Shahnaz","year":"2006","unstructured":"Shahnaz, F., Berry, M.W., Pauca, V.P., Plemmons, R.J.: Document clustering using nonnegative matrix factorization. Inf. Process. Manag. 42, 373\u2013386 (2006)","journal-title":"Inf. Process. Manag."},{"key":"9507_CR28","first-page":"1457","volume":"5","author":"P.O. Hoyer","year":"2004","unstructured":"Hoyer, P.O.: Non-negative matrix factorization with sparseness constraints. J. Mach. Learn. Res. 5, 1457\u20131469 (2004)","journal-title":"J. Mach. Learn. Res."},{"key":"9507_CR29","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/S0167-6377(99)00074-7","volume":"26","author":"L. Grippo","year":"2000","unstructured":"Grippo, L., Sciandrone, M.: On the convergence of the block nonlinear Gauss-Seidel method under convex constraints. Oper. Res. Lett. 26, 127\u2013136 (2000)","journal-title":"Oper. Res. Lett."},{"key":"9507_CR30","doi-asserted-by":"crossref","first-page":"870","DOI":"10.1007\/11785231_91","volume-title":"International Conference on Artificial Intelligence and Soft Computing (ICAISC 2006)","author":"R. Zdunek","year":"2006","unstructured":"Zdunek, R., Cichocki, A.: Non-negative matrix factorization with quasi-newton optimization. In: International Conference on Artificial Intelligence and Soft Computing (ICAISC 2006), pp. 870\u2013879 (2006)"},{"key":"9507_CR31","volume-title":"Proceedings of the Workshop on Text Mining at the 6th SIAM International Conference on Data Mining (SDM06)","author":"H. Park","year":"2006","unstructured":"Park, H., Kim, H.: One-sided non-negative matrix factorization and non-negative centroid dimension reduction for text classification. In: Proceedings of the Workshop on Text Mining at the 6th SIAM International Conference on Data Mining (SDM06) (2006)"},{"key":"9507_CR32","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s00245-008-9038-9","volume":"58","author":"Y. Xiao","year":"2008","unstructured":"Xiao, Y., Hu, Q.: Subspace Barzilai-Borwein gradient method for large-scale bound constrained optimization. Appl. Math. Optim. 58, 275\u2013290 (2008)","journal-title":"Appl. Math. Optim."},{"key":"9507_CR33","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":"9507_CR34","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1007\/s10107-004-0516-9","volume":"103","author":"Y.-H. Dai","year":"2005","unstructured":"Dai, Y.-H., Fletcher, R.: On the asymptotic behaviour of some new gradient methods. Math. Program. Ser. A, B 103, 541\u2013559 (2005)","journal-title":"Math. Program. Ser. A, B"},{"key":"9507_CR35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/imanum\/22.1.1","volume":"22","author":"Y.-H. Dai","year":"2002","unstructured":"Dai, Y.-H., Liao, L.-Z.: R-linear convergence of the Barzilai and Borwein gradient method. IMA J. Numer. Anal. 22, 1\u201310 (2002)","journal-title":"IMA J. Numer. Anal."},{"key":"9507_CR36","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1023\/A:1014838419611","volume":"22","author":"Y. Dai","year":"2002","unstructured":"Dai, Y., Yuan, J., Yuan, Y.-X.: Modified two-point stepsize gradient methods for unconstrained optimization. Comput. Optim. Appl. 22, 103\u2013109 (2002)","journal-title":"Comput. Optim. Appl."},{"key":"9507_CR37","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1137\/S1052623497330963","volume":"10","author":"E.G. Birgin","year":"2000","unstructured":"Birgin, E.G., Mart\u00ednez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Control Optim. 10, 1196\u20131211 (2000)","journal-title":"SIAM J. Control Optim."},{"key":"9507_CR38","doi-asserted-by":"crossref","first-page":"299","DOI":"10.3934\/jimo.2008.4.299","volume":"4","author":"G. Frassoldati","year":"2008","unstructured":"Frassoldati, G., Zanni, L., Zanghirati, G.: New adaptive stepsize selections in gradient methods. J. Ind. Manag. Optim. 4, 299\u2013312 (2008)","journal-title":"J. Ind. Manag. Optim."},{"key":"9507_CR39","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s00211-004-0569-y","volume":"100","author":"Y.-H. Dai","year":"2005","unstructured":"Dai, Y.-H., Fletcher, R.: Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. Numer. Math. 100, 21\u201347 (2005)","journal-title":"Numer. Math."},{"key":"9507_CR40","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1137\/0723046","volume":"23","author":"L. Grippo","year":"1986","unstructured":"Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton\u2019s method. SIAM J. Numer. Anal. 23, 707\u2013716 (1986)","journal-title":"SIAM J. Numer. Anal."},{"key":"9507_CR41","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1137\/S1052623403428208","volume":"14","author":"H. Zhang","year":"2004","unstructured":"Zhang, H., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Control Optim. 14, 1043\u20131056 (2004)","journal-title":"SIAM J. Control Optim."},{"key":"9507_CR42","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1137\/S106482759427021X","volume":"17","author":"P.L. Toint","year":"1996","unstructured":"Toint, P.L.: An assessment of nonmonotone linesearch techniques for unconstrained optimization. SIAM J. Sci. Comput. 17, 725\u2013739 (1996)","journal-title":"SIAM J. Sci. Comput."},{"key":"9507_CR43","doi-asserted-by":"crossref","first-page":"936","DOI":"10.1137\/0322061","volume":"22","author":"E.M. Gafni","year":"1984","unstructured":"Gafni, E.M., Bertsekas, D.P.: Two-metric projection methods for constrained optimization. SIAM J. Control Optim. 22, 936\u2013964 (1984)","journal-title":"SIAM J. Control Optim."},{"key":"9507_CR44","unstructured":"Zhang, Y.: An alternating direction algorithm for nonnegative matrix factorization. Technical report, Rice University (2010)"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-012-9507-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-012-9507-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-012-9507-6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T11:36:43Z","timestamp":1559302603000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-012-9507-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9,26]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2013,5]]}},"alternative-id":["9507"],"URL":"https:\/\/doi.org\/10.1007\/s10589-012-9507-6","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,9,26]]}}}