{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T06:29:22Z","timestamp":1768976962776,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T00:00:00Z","timestamp":1706486400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201492"],"award-info":[{"award-number":["12201492"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we first develop an active set identification technique, and then we suggest a modified nonmonotone line search rule, in which a new parameter formula is introduced to control the degree of the nonmonotonicity of line search. By using the modified line search and the active set identification technique, we propose a global convergent method to solve the NMF based on the alternating nonnegative least squares framework. In addition, the larger step size technique is exploited to accelerate convergence. Finally, a large number of numerical experiments are carried out on synthetic and image datasets, and the results show that our presented method is effective in calculating speed and solution quality.<\/jats:p>","DOI":"10.3390\/sym16020154","type":"journal-article","created":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T04:57:07Z","timestamp":1706504227000},"page":"154","update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Gradient-Based Algorithm with Nonmonotone Line Search for Nonnegative Matrix Factorization"],"prefix":"10.3390","volume":"16","author":[{"given":"Wenbo","family":"Li","sequence":"first","affiliation":[{"name":"School of Sciences, Xi\u2019an University of Technology, Xi\u2019an 710054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaolu","family":"Shi","sequence":"additional","affiliation":[{"name":"Beijing Mechanical Equipment Institute, Beijing 100039, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3557","DOI":"10.1016\/j.patcog.2012.02.037","article-title":"Efficient nonnegative matrix factorization via projected Newton method","volume":"45","author":"Gong","year":"2012","journal-title":"Pattern Recognit."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1038\/44565","article-title":"Learning the parts of objects by non-negative matrix factorization","volume":"401","author":"Lee","year":"1999","journal-title":"Nature"},{"key":"ref_3","first-page":"556","article-title":"Algorithms for non-negative matrix factorization","volume":"13","author":"Lee","year":"2001","journal-title":"Adv. Neural Process. Inf. Syst."},{"key":"ref_4","first-page":"38","article-title":"Fast Newton-type methods for the least squares nonnegative matrix approximation problem","volume":"1","author":"Kim","year":"2007","journal-title":"SIAM Int. Conf. Data Min."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1002\/env.3170050203","article-title":"Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values","volume":"5","author":"Paatero","year":"1994","journal-title":"Environmetrics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3913","DOI":"10.1016\/j.csda.2008.01.011","article-title":"On the equivalence between non-negative matrix factorization and probabilistic latent semantic indexing","volume":"52","author":"Ding","year":"2008","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5120","DOI":"10.1109\/TSP.2008.928937","article-title":"A convex analysis framework for blind separation of nonnegative sources","volume":"56","author":"Chan","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ding, C., He, X., and Simon, H. (2005). On the Equivalence of Nonnegative Matrix Factorization and Spectral Clustering. SIAM Int. Conf. Data Min. (SDM\u201905), 606\u2013610.","DOI":"10.1137\/1.9781611972757.70"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1162\/neco.2008.04-08-771","article-title":"Nonnegative matrix factorization with the Itakura-Saito divergence: With application to music analysis","volume":"21","author":"Bertin","year":"2009","journal-title":"Neural Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1109\/MSP.2013.2279731","article-title":"A Signal Processing Perspective on Hyperspectral Unmixing","volume":"31","author":"Ma","year":"2014","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1093\/imanum\/8.1.141","article-title":"Two-point step size gradient methods","volume":"8","author":"Barzilai","year":"1988","journal-title":"IMA J. Numer. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/imanum\/22.1.1","article-title":"R-Linear convergence of the Barzilai-Borwein gradient method","volume":"22","author":"Dai","year":"2002","journal-title":"IMA J. Numer. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1093\/imanum\/13.3.321","article-title":"On the Barzilai-Borwein choice of steplength for the gradient method","volume":"13","author":"Raydan","year":"1993","journal-title":"IMA J. Numer. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1137\/S1052623494266365","article-title":"The Barzilai and Borwein gradient method for the large-scale unconstrained minimization problem","volume":"7","author":"Raydan","year":"1997","journal-title":"SIAM J. Optim."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s00245-008-9038-9","article-title":"Subspace Barzilai-Borwein gradient method for large-scale bound constrained optimization","volume":"58","author":"Xiao","year":"2008","journal-title":"Appl. Math. Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3117","DOI":"10.1016\/j.apm.2010.09.011","article-title":"Modified active set projected spectral gradient method for bound constrained optimization","volume":"35","author":"Xiao","year":"2011","journal-title":"Appl. Math. Model."},{"key":"ref_17","first-page":"54","article-title":"Alternating projected Barzilai-Borwein methods for nonnegative matrix factorization","volume":"36","author":"Han","year":"2009","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1665","DOI":"10.1007\/s10618-014-0390-x","article-title":"Quadratic regularization projected alternating Barzilai-Borwein method for nonnegative matrix factorization","volume":"29","author":"Huang","year":"2015","journal-title":"Data Min. Knowl. Discov."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.aml.2015.01.003","article-title":"An efficint monotone projected Barzilai-Borwein method for nonnegative matrix factorization","volume":"45","author":"Huang","year":"2015","journal-title":"Appl. Math. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1007\/s10589-010-9387-6","article-title":"Non-monotone projection gradient method for non-negative matrix factorization","volume":"51","author":"Li","year":"2012","journal-title":"Comput. Optim. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s10589-012-9507-6","article-title":"Modified subspace Barzilai-Borwein gradient method for non-negative matrix factorization","volume":"55","author":"Liu","year":"2013","journal-title":"Comput. Optim. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1431","DOI":"10.1093\/imanum\/drq024","article-title":"Inexact block coordinate descent methods with application to non-negative matrix factorization","volume":"31","author":"Bonettini","year":"2011","journal-title":"IMA J. Numer. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"939567","DOI":"10.1155\/2008\/939567","article-title":"Fast nonnegative matrix factorization algorithms using projected gradient approaches for large-scale problems","volume":"2008","author":"Zdunek","year":"2008","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1007\/s40305-023-00470-8","article-title":"Accelerated stochastic Peaceman-Rachford method for empirical risk minimization","volume":"11","author":"Bai","year":"2023","journal-title":"J. Oper. Res. Soc. China"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"448","DOI":"10.4208\/csiam-am.SO-2021-0021","article-title":"Convergence on a symmetric accelerated stochastic ADMM with larger stepsizes","volume":"3","author":"Bai","year":"2022","journal-title":"CSIAM Trans. Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s10589-021-00338-8","article-title":"An inexact accelerated stochastic ADMM for separable convex optimization","volume":"81","author":"Bai","year":"2022","journal-title":"Comput. Optim. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s10589-017-9971-0","article-title":"Generalized symmetric ADMM for separable convex optimization","volume":"70","author":"Bai","year":"2018","journal-title":"Comput. Optim. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1007\/s11590-017-1195-9","article-title":"A parameterized proximal point algorithm for separable convex optimization","volume":"12","author":"Bai","year":"2018","journal-title":"Optim. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2882","DOI":"10.1109\/TSP.2012.2190406","article-title":"NeNMF: An optimal gradient method for nonnegative matrix factorization","volume":"60","author":"Guan","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1007\/s10915-017-0376-0","article-title":"A globally convergent algorithm for nonconvex optimization based on block coordinate update","volume":"72","author":"Xu","year":"2017","journal-title":"J. Sci. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1137\/S1052623403428208","article-title":"A nonmonotone line search technique and its application to unconstrained optimization","volume":"14","author":"Zhang","year":"2004","journal-title":"SIAM J. Optim."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1023\/A:1013653923062","article-title":"On the nonmonotone line search","volume":"112","author":"Dai","year":"2002","journal-title":"J. Optim. Theory Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2158","DOI":"10.1016\/j.camwa.2007.08.038","article-title":"Incorporating nonmonotone strategies into the trust region method for unconstrained optimization","volume":"55","author":"Gu","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1080\/02331934.2011.641126","article-title":"A class of nonmonotone Armijo-type line search method for unconstrained optimization","volume":"61","author":"Ahookhosh","year":"2012","journal-title":"Optimization"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1007\/s10092-017-0226-3","article-title":"On the nonmonotonicity degree of nonmonotone line searches","volume":"54","author":"Nosratipour","year":"2017","journal-title":"Calcolo"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Glowinski, R. (1984). Numerical Methods for Nonlinear Variational Problems, Springer.","DOI":"10.1007\/978-3-662-12613-4"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1137\/S1052623497330963","article-title":"Nonmonotone spectral projected gradient methods on convex sets","volume":"10","author":"Birgin","year":"2000","journal-title":"SIAM J. Optim."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/978-3-540-74494-8_22","article-title":"Hierarchical ALS Algorithms for Nonnegative Matrix and 3D Tensor Factorization","volume":"4666","author":"Cichocki","year":"2007","journal-title":"Lect. Notes Comput. Sci. Springer"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1758","DOI":"10.1137\/120887795","article-title":"A block coordinate descent method for regularized multi-convex optimization with applications to nonnegative tensor factorization and completion","volume":"6","author":"Xu","year":"2015","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2756","DOI":"10.1162\/neco.2007.19.10.2756","article-title":"Projected Gradient Methods for non-negative matrix factorization","volume":"19","author":"Lin","year":"2007","journal-title":"Neural Comput."},{"key":"ref_41","unstructured":"Gillis, N. (2015). The why and how of nonnegative matrix factorization. arXiv."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.mdpi.com\/2073-8994\/16\/2\/154\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:50:51Z","timestamp":1760104251000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.mdpi.com\/2073-8994\/16\/2\/154"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,29]]},"references-count":41,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["sym16020154"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.3390\/sym16020154","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,29]]}}}