{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T09:50:53Z","timestamp":1784713853524,"version":"3.55.0"},"reference-count":39,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T00:00:00Z","timestamp":1685664000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Nature Science Foundation of China","award":["62171404"],"award-info":[{"award-number":["62171404"]}]},{"name":"National Nature Science Foundation of China","award":["8091B022120"],"award-info":[{"award-number":["8091B022120"]}]},{"name":"National Nature Science Foundation of China","award":["LQ23F020003"],"award-info":[{"award-number":["LQ23F020003"]}]},{"name":"National Nature Science Foundation of China","award":["G20220007"],"award-info":[{"award-number":["G20220007"]}]},{"name":"Joint Fund of the Ministry of Education of China","award":["62171404"],"award-info":[{"award-number":["62171404"]}]},{"name":"Joint Fund of the Ministry of Education of China","award":["8091B022120"],"award-info":[{"award-number":["8091B022120"]}]},{"name":"Joint Fund of the Ministry of Education of China","award":["LQ23F020003"],"award-info":[{"award-number":["LQ23F020003"]}]},{"name":"Joint Fund of the Ministry of Education of China","award":["G20220007"],"award-info":[{"award-number":["G20220007"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["62171404"],"award-info":[{"award-number":["62171404"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["8091B022120"],"award-info":[{"award-number":["8091B022120"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LQ23F020003"],"award-info":[{"award-number":["LQ23F020003"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["G20220007"],"award-info":[{"award-number":["G20220007"]}]},{"name":"Natural Science Foundation of Wenzhou","award":["62171404"],"award-info":[{"award-number":["62171404"]}]},{"name":"Natural Science Foundation of Wenzhou","award":["8091B022120"],"award-info":[{"award-number":["8091B022120"]}]},{"name":"Natural Science Foundation of Wenzhou","award":["LQ23F020003"],"award-info":[{"award-number":["LQ23F020003"]}]},{"name":"Natural Science Foundation of Wenzhou","award":["G20220007"],"award-info":[{"award-number":["G20220007"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>Nonnegative matrix factorization (NMF) and its numerous variants have been extensively studied and used in hyperspectral unmixing (HU). With the aid of the designed deep structure, deep NMF-based methods demonstrate advantages in exploring the hierarchical features of complex data. However, a noise corruption problem commonly exists in hyperspectral data and severely degrades the unmixing performance of deep NMF-based methods when applied to HU. In this study, we propose an \u21132,1 norm-based robust deep nonnegative matrix factorization (\u21132,1-RDNMF) for HU, which incorporates an \u21132,1 norm into the two stages of the deep structure to achieve robustness. The multiplicative updating rules of \u21132,1-RDNMF are efficiently learned and provided. The efficiency of the presented method is verified in experiments using both synthetic and genuine data.<\/jats:p>","DOI":"10.3390\/rs15112900","type":"journal-article","created":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T08:50:31Z","timestamp":1685695831000},"page":"2900","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Hyperspectral Unmixing Using Robust Deep Nonnegative Matrix Factorization"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6661-023X","authenticated-orcid":false,"given":"Risheng","family":"Huang","sequence":"first","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Shaoxing University, Shaoxing 312000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huiyun","family":"Jiao","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaorun","family":"Li","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuhan","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0167-0423","authenticated-orcid":false,"given":"Chaoqun","family":"Xia","sequence":"additional","affiliation":[{"name":"College of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou 325035, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4414","DOI":"10.1109\/JSTARS.2022.3175257","article-title":"Hyperspectral unmixing based on nonnegative matrix factorization: A comprehensive review","volume":"15","author":"Feng","year":"2022","journal-title":"IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4282","DOI":"10.1109\/TGRS.2011.2144605","article-title":"Hyperspectral unmixing via L1\/2 sparsity-constrained nonnegative matrix factorization","volume":"49","author":"Qian","year":"2011","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2590","DOI":"10.1109\/TGRS.2009.2038483","article-title":"Minimum dispersion constrained nonnegative matrix factorization to unmix hyperspectral data","volume":"48","author":"Huck","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2815","DOI":"10.1109\/TGRS.2012.2213825","article-title":"Manifold regularized sparse NMF for hyperspectral unmixing","volume":"51","author":"Lu","year":"2013","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1109\/JSTARS.2013.2242255","article-title":"An endmember dissimilarity constrained non-negative matrix factorization method for hyperspectral unmixing","volume":"6","author":"Wang","year":"2013","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"6531","DOI":"10.1109\/TGRS.2016.2586110","article-title":"Nonnegative-matrix-factorization-based hyperspectral unmixing with partially known endmembers","volume":"54","author":"Lei","year":"2016","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4810","DOI":"10.1109\/TIP.2015.2468177","article-title":"Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization","volume":"24","author":"Dobigeon","year":"2015","journal-title":"IEEE Trans. Image Process."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Li, C., Ma, Y., Mei, X., Liu, C., and Ma, J. (2016). Hyperspectral unmixing with robust collaborative sparse regression. Remote Sens., 8.","DOI":"10.3390\/rs8070588"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4267","DOI":"10.1109\/JSTARS.2016.2519498","article-title":"Sparsity-regularized robust non-negative matrix factorization for hyperspectral unmixing","volume":"9","author":"He","year":"2016","journal-title":"IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens."},{"key":"ref_10","unstructured":"Ding, C., Zhou, D., He, X., and Zha, H. (2006, January 25\u201329). R1-PCA: Rotational invariant L1-norm principal component analysis for robust subspace factorization. Proceedings of the 23rd International Conference on Machine Learning, Pittsburgh, PA, USA."},{"key":"ref_11","unstructured":"Nie, F., Huang, H., Cai, X., and Ding, C.H. (2010). Advances in Neural Information Processing Systems 23 (NIPS 2010), Morgan Kaufmann."},{"key":"ref_12","unstructured":"Huang, H., and Ding, C. (2008, January 23\u201328). Robust tensor factorization using R1 norm. Proceedings of the 2008 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Anchorage, AK, USA."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2708","DOI":"10.1016\/j.patcog.2012.01.003","article-title":"Robust classification using \u21132,1-norm based regression model","volume":"45","author":"Ren","year":"2012","journal-title":"Pattern Recognit."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1007\/s00521-013-1371-5","article-title":"Robust non-negative matrix factorization via joint sparse and graph regularization for transfer learning","volume":"23","author":"Yang","year":"2013","journal-title":"Neur. Comput. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1109\/TGRS.2016.2616161","article-title":"Robust sparse hyperspectral unmixing with \u21132,1 norm","volume":"55","author":"Ma","year":"2017","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kong, D., Ding, C., and Huang, H. (2011, January 24\u201328). Robust nonnegative matrix factorization using L21-norm. Proceedings of the 20th ACM International Conference on Information and Knowledge Management, Glasgow, UK.","DOI":"10.1145\/2063576.2063676"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4027","DOI":"10.1109\/TIP.2015.2456508","article-title":"Robust hyperspectral unmixing with correntropy-based metric","volume":"24","author":"Wang","year":"2015","journal-title":"IEEE Trans. Image Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1109\/TGRS.2020.2999936","article-title":"Correntropy-based spatial-spectral robust sparsity-regularized hyperspectral unmixing","volume":"59","author":"Li","year":"2020","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"104898","DOI":"10.1016\/j.knosys.2019.104898","article-title":"Cauchy sparse NMF with manifold regularization: A robust method for hyperspectral unmixing","volume":"184","author":"Wang","year":"2019","journal-title":"Knowl.-Based Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"8235","DOI":"10.1109\/TGRS.2019.2919166","article-title":"Spectral-spatial robust nonnegative matrix factorization for hyperspectral unmixing","volume":"57","author":"Huang","year":"2019","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1501","DOI":"10.1109\/TGRS.2020.2996688","article-title":"Self-paced nonnegative matrix factorization for hyperspectral unmixing","volume":"59","author":"Peng","year":"2021","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7723","DOI":"10.1007\/s00521-020-05514-1","article-title":"Spectrum interference-based two-level data augmentation method in deep learning for automatic modulation classification","volume":"33","author":"Zheng","year":"2021","journal-title":"Neur. Comput. Appl."},{"key":"ref_23","unstructured":"Zhao, M., Liu, Q., Jha, A., Deng, R., Yao, T., Mahadevan-Jansen, A., Tyska, M.J., Millis, B.A., and Huo, Y. (2021). Proceedings of the Machine Learning in Medical Imaging: 12th International Workshop, MLMI 2021, Held in Conjunction with MICCAI 2021, Strasbourg, France, 27 September 2021, Proceedings 12, Springer."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1109\/LGRS.2014.2325874","article-title":"Spectral unmixing of hyperspectral imagery using multilayer NMF","volume":"12","author":"Rajabi","year":"2014","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1049\/el:20060983","article-title":"Multilayer nonnegative matrix factorisation","volume":"42","author":"Cichocki","year":"2006","journal-title":"Electron. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"100423","DOI":"10.1016\/j.cosrev.2021.100423","article-title":"A survey on deep matrix factorizations","volume":"42","author":"Gillis","year":"2021","journal-title":"Comput. Sci. Rev."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1109\/TPAMI.2016.2554555","article-title":"A deep matrix factorization method for learning attribute representations","volume":"39","author":"Trigeorgis","year":"2016","journal-title":"IEEE Trans. Patt. Anal. Mach. Intell."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.neucom.2021.08.152","article-title":"A survey of deep nonnegative matrix factorization","volume":"491","author":"Chen","year":"2022","journal-title":"Neurocomputing"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"109210","DOI":"10.1016\/j.knosys.2022.109210","article-title":"Deep alternating non-negative matrix factorisation","volume":"251","author":"Sun","year":"2022","journal-title":"Knowl.-Based Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1109\/LGRS.2018.2823425","article-title":"Sparsity-constrained deep nonnegative matrix factorization for hyperspectral unmixing","volume":"15","author":"Fang","year":"2018","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"6245","DOI":"10.1109\/TGRS.2018.2834567","article-title":"Hyperspectral unmixing using sparsity-constrained deep nonnegative matrix factorization with total variation","volume":"56","author":"Feng","year":"2018","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Huang, R., Li, X., Fang, Y., Cao, Z., and Xia, C. (2023). Robust Hyperspectral Unmixing with Practical Learning-Based Hyperspectral Image Denoising. Remote Sens., 15.","DOI":"10.3390\/rs15041058"},{"key":"ref_33","first-page":"1","article-title":"Correntropy-based autoencoder-like NMF with total variation for hyperspectral unmixing","volume":"19","author":"Feng","year":"2022","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_34","unstructured":"Lee, D.D., and Seung, H.S. (2001). Advances in Neural Information Processing Systems 13 (NIPS 2000), Morgan Kaufmann."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"898","DOI":"10.1109\/TGRS.2005.844293","article-title":"Vertex component analysis: A fast algorithm to unmix hyperspectral data","volume":"43","author":"Nascimento","year":"2005","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1109\/36.911111","article-title":"Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery","volume":"39","author":"Heinz","year":"2001","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Clark, R.N., Swayze, G.A., Gallagher, A.J., King, T.V., and Calvin, W.M. (1993). The US Geological Survey, Digital Spectral Library: Version 1 (0.2 to 3.0 um), U.S. Geological Survey Open-File ReportThe US Geological Survey.","DOI":"10.3133\/ofr93592"},{"key":"ref_38","unstructured":"Zhu, F. (2019, March 10). Hyperspectral Unmixing Datasets & Ground Truths. Available online: http:\/\/www.escience.cn\/people\/feiyunZHU\/Dataset_GT.html."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"5412","DOI":"10.1109\/TIP.2014.2363423","article-title":"Spectral unmixing via data-guided sparsity","volume":"23","author":"Zhu","year":"2014","journal-title":"IEEE Trans. Image Process."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/11\/2900\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:47:56Z","timestamp":1760125676000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/11\/2900"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,2]]},"references-count":39,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["rs15112900"],"URL":"https:\/\/doi.org\/10.3390\/rs15112900","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,2]]}}}