{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T02:08:06Z","timestamp":1780366086915,"version":"3.54.1"},"reference-count":40,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,12,1]],"date-time":"2026-12-01T00:00:00Z","timestamp":1796083200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Signal Processing"],"published-print":{"date-parts":[[2026,12]]},"DOI":"10.1016\/j.sigpro.2026.110723","type":"journal-article","created":{"date-parts":[[2026,5,28]],"date-time":"2026-05-28T15:02:03Z","timestamp":1779980523000},"page":"110723","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Deep unfolded maximum correntropy network: A trainable framework for adaptive filtering"],"prefix":"10.1016","volume":"249","author":[{"given":"Xinyan","family":"Hou","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jun","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haiquan","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoqiang","family":"Long","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-3153-3604","authenticated-orcid":false,"given":"Bing","family":"Ren","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"3","key":"10.1016\/j.sigpro.2026.110723_bib0001","first-page":"62","article-title":"Adaptive equalization algorithms: an overview","volume":"2","author":"Malik","year":"2011","journal-title":"Int. J. Adv. Comput. Sci. Appl."},{"key":"10.1016\/j.sigpro.2026.110723_bib0002","series-title":"Fundamentals of Adaptive Filtering","author":"Sayed","year":"2003"},{"issue":"2","key":"10.1016\/j.sigpro.2026.110723_bib0003","doi-asserted-by":"crossref","first-page":"2567","DOI":"10.1109\/TCE.2025.3570854","article-title":"Robust Volterra filter for nonlinear censored regression and its applications to nonlinear acoustic echo cancellation","volume":"71","author":"Qian","year":"2025","journal-title":"IEEE Trans. Consum. Electron."},{"key":"10.1016\/j.sigpro.2026.110723_bib0004","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1109\/TASLP.2023.3330077","article-title":"Error reused filtered-X least mean square algorithm for active noise control","volume":"32","author":"Zhang","year":"2024","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"issue":"7","key":"10.1016\/j.sigpro.2026.110723_bib0005","first-page":"3588","article-title":"A robust DOA estimator based on maximum mixture complex correntropy for noisy input and impulsive noise","volume":"71","author":"Zhao","year":"2024","journal-title":"IEEE Trans. Circuits Syst. II: Express Br."},{"issue":"9","key":"10.1016\/j.sigpro.2026.110723_bib0006","doi-asserted-by":"crossref","first-page":"2811","DOI":"10.1109\/78.236504","article-title":"On the convergence behavior of the LMS and the normalized LMS algorithms","volume":"41","author":"Slock","year":"1993","journal-title":"IEEE Trans. Signal Process."},{"issue":"1","key":"10.1016\/j.sigpro.2026.110723_bib0007","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1109\/TSP.2009.2025074","article-title":"Distributed estimation over an adaptive incremental network based on the affine projection algorithm","volume":"58","author":"Li","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"issue":"3","key":"10.1016\/j.sigpro.2026.110723_bib0008","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1109\/MCS.2019.2900788","article-title":"Recursive least squares for real-time implementation [lecture notes]","volume":"39","author":"Islam","year":"2019","journal-title":"IEEE Control Syst. Mag."},{"key":"10.1016\/j.sigpro.2026.110723_bib0009","series-title":"Adaptive Filters","author":"Sayed","year":"2011"},{"issue":"2","key":"10.1016\/j.sigpro.2026.110723_bib0010","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1109\/LSP.2003.821722","article-title":"Variable step-size NLMS and affine projection algorithms","volume":"11","author":"Shin","year":"2004","journal-title":"IEEE Signal Process. Lett."},{"issue":"5","key":"10.1016\/j.sigpro.2026.110723_bib0011","doi-asserted-by":"crossref","first-page":"1878","DOI":"10.1109\/TSP.2007.913142","article-title":"A new robust variable step-size NLMS algorithm","volume":"56","author":"Vega","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"10.1016\/j.sigpro.2026.110723_bib0012","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1109\/LSP.2008.2001559","article-title":"A robust variable forgetting factor recursive least-squares algorithm for system identification","volume":"15","author":"Paleologu","year":"2008","journal-title":"IEEE Signal Process. Lett."},{"issue":"5","key":"10.1016\/j.sigpro.2026.110723_bib0013","doi-asserted-by":"crossref","first-page":"2353","DOI":"10.1109\/TPWRD.2015.2422139","article-title":"Variable forgetting factor recursive least square control algorithm for DSTATCOM","volume":"30","author":"Badoni","year":"2015","journal-title":"IEEE Trans. Power Deliv."},{"key":"10.1016\/j.sigpro.2026.110723_bib0014","series-title":"Information Theoretic Learning: Renyi\u2019s Entropy and Kernel Perspectives","author":"Principe","year":"2010"},{"key":"10.1016\/j.sigpro.2026.110723_bib0015","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.eswa.2018.04.020","article-title":"Complex correntropy function: properties, and application to a channel equalization problem","volume":"107","author":"Guimar\u00e3es","year":"2018","journal-title":"Expert Syst. Appl."},{"issue":"11","key":"10.1016\/j.sigpro.2026.110723_bib0016","doi-asserted-by":"crossref","first-page":"5286","DOI":"10.1109\/TSP.2007.896065","article-title":"Correntropy: properties and applications in non-Gaussian signal processing","volume":"55","author":"Liu","year":"2007","journal-title":"IEEE Trans. Signal Process."},{"issue":"6","key":"10.1016\/j.sigpro.2026.110723_bib0017","doi-asserted-by":"crossref","first-page":"4007","DOI":"10.1109\/TSMC.2019.2931403","article-title":"Effects of outliers on the maximum correntropy estimation: a robustness analysis","volume":"51","author":"Chen","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybern.: Syst."},{"issue":"2","key":"10.1016\/j.sigpro.2026.110723_bib0018","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1109\/TNNLS.2020.3029198","article-title":"Mixture correntropy-based kernel extreme learning machines","volume":"33","author":"Zheng","year":"2022","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"3","key":"10.1016\/j.sigpro.2026.110723_bib0019","first-page":"1438","article-title":"Gauss hermite fourier features based on maximum correntropy criterion for adaptive filtering","volume":"72","author":"Zhou","year":"2025","journal-title":"IEEE Trans. Circuits Syst. I: Regul. Pap."},{"issue":"7","key":"10.1016\/j.sigpro.2026.110723_bib0020","doi-asserted-by":"crossref","first-page":"3083","DOI":"10.1109\/TNNLS.2020.3009417","article-title":"Broad learning system based on maximum correntropy criterion","volume":"32","author":"Zheng","year":"2021","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"11","key":"10.1016\/j.sigpro.2026.110723_bib0021","doi-asserted-by":"crossref","first-page":"19964","DOI":"10.1109\/TNNLS.2025.3590097","article-title":"Robust fault-aware extreme learning machine based on maximum correntropy","volume":"36","author":"Xiao","year":"2025","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"2","key":"10.1016\/j.sigpro.2026.110723_bib0022","first-page":"604","article-title":"Robust maximum correntropy criterion subband adaptive filter algorithm for impulsive noise and noisy input","volume":"69","author":"Zhao","year":"2021","journal-title":"IEEE Trans. Circuits Syst. II: Express Br."},{"key":"10.1016\/j.sigpro.2026.110723_bib0023","doi-asserted-by":"crossref","DOI":"10.1016\/j.sigpro.2020.107589","article-title":"Statistics variable kernel width for maximum correntropy criterion algorithm","volume":"176","author":"Zhou","year":"2020","journal-title":"Signal Process."},{"issue":"7","key":"10.1016\/j.sigpro.2026.110723_bib0024","first-page":"1339","article-title":"An improved variable kernel width for maximum correntropy criterion algorithm","volume":"67","author":"Shi","year":"2018","journal-title":"IEEE Trans. Circuits Syst. II: Express Br."},{"key":"10.1016\/j.sigpro.2026.110723_bib0025","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1109\/LSP.2023.3273174","article-title":"An efficient parameter optimization of maximum correntropy criterion","volume":"30","author":"Shi","year":"2023","journal-title":"IEEE Signal Process. Lett."},{"issue":"10","key":"10.1016\/j.sigpro.2026.110723_bib0026","first-page":"1247","article-title":"Adaptive filtering under a variable kernel width maximum correntropy criterion","volume":"64","author":"Huang","year":"2017","journal-title":"IEEE Trans. Circuits Syst. II: Express Br."},{"issue":"5","key":"10.1016\/j.sigpro.2026.110723_bib0027","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1109\/JPROC.2023.3247480","article-title":"Model-based deep learning","volume":"111","author":"Shlezinger","year":"2023","journal-title":"Proc. IEEE"},{"key":"10.1016\/j.sigpro.2026.110723_bib0028","doi-asserted-by":"crossref","DOI":"10.1016\/j.eswa.2023.119807","article-title":"Adaptive weighted rain streaks model-driven deep network for single image deraining","volume":"222","author":"Zhang","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"10.1016\/j.sigpro.2026.110723_bib0029","series-title":"Proceedings of the 27th International Conference on International Conference on Machine Learning","first-page":"399","article-title":"Learning fast approximations of sparse coding","author":"Gregor","year":"2010"},{"issue":"3","key":"10.1016\/j.sigpro.2026.110723_bib0030","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1109\/TPAMI.2018.2883941","article-title":"ADMM-CSNet: a deep learning approach for image compressive sensing","volume":"42","author":"Yang","year":"2018","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"10.1016\/j.sigpro.2026.110723_bib0031","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1109\/TCI.2020.2964202","article-title":"Efficient and interpretable deep blind image deblurring via algorithm unrolling","volume":"6","author":"Li","year":"2020","journal-title":"IEEE Trans. Comput. Imaging"},{"key":"10.1016\/j.sigpro.2026.110723_bib0032","doi-asserted-by":"crossref","DOI":"10.1016\/j.eswa.2025.131019","article-title":"Edge priors guided deep unrolling network for single image super-resolution","volume":"308","author":"Song","year":"2026","journal-title":"Expert Syst. Appl."},{"issue":"10","key":"10.1016\/j.sigpro.2026.110723_bib0033","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1109\/TSP.2019.2899805","article-title":"Learning to detect","volume":"67","author":"Samuel","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"10.1016\/j.sigpro.2026.110723_bib0034","series-title":"Proceedings of the 34th Annual International Symposium on Personal, Indoor and Mobile Radio","first-page":"2166","article-title":"Deep learning based DOA estimation with trainable-step-size LMS algorithm","author":"Guo","year":"2023"},{"key":"10.1016\/j.sigpro.2026.110723_bib0035","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.1109\/LSP.2022.3192207","article-title":"An innovative transient analysis of adaptive filter with maximum correntropy criterion","volume":"29","author":"Hou","year":"2022","journal-title":"IEEE Signal Process. Lett."},{"issue":"7","key":"10.1016\/j.sigpro.2026.110723_bib0036","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1109\/LSP.2014.2319308","article-title":"Steady-state mean-square error analysis for adaptive filtering under the maximum correntropy criterion","volume":"21","author":"Chen","year":"2014","journal-title":"IEEE Signal Process. Lett."},{"key":"10.1016\/j.sigpro.2026.110723_bib0037","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.sigpro.2017.05.029","article-title":"Maximum total correntropy adaptive filtering against heavy-tailed noises","volume":"141","author":"Wang","year":"2017","journal-title":"Signal Process."},{"issue":"10","key":"10.1016\/j.sigpro.2026.110723_bib0038","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1002\/tee.23217","article-title":"A variable step size for maximum correntropy criterion algorithm with improved variable kernel width","volume":"15","author":"Wang","year":"2020","journal-title":"IEEJ Trans. Electr. Electron. Eng."},{"issue":"8","key":"10.1016\/j.sigpro.2026.110723_bib0039","doi-asserted-by":"crossref","first-page":"1576","DOI":"10.1109\/LAWP.2019.2923700","article-title":"Efficient direction-of-arrival estimation method based on variable-step-size LMS algorithm","volume":"18","author":"Jalal","year":"2019","journal-title":"IEEE Antennas Wirel. Propag. Lett."},{"issue":"12","key":"10.1016\/j.sigpro.2026.110723_bib0040","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1109\/CC.2016.7897554","article-title":"DOA estimation algorithm based on adaptive filtering in spatial domain","volume":"13","author":"Zeng","year":"2016","journal-title":"China Commun."}],"container-title":["Signal Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0165168426002379?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0165168426002379?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T01:59:49Z","timestamp":1780365589000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0165168426002379"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,12]]},"references-count":40,"alternative-id":["S0165168426002379"],"URL":"https:\/\/doi.org\/10.1016\/j.sigpro.2026.110723","relation":{},"ISSN":["0165-1684"],"issn-type":[{"value":"0165-1684","type":"print"}],"subject":[],"published":{"date-parts":[[2026,12]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Deep unfolded maximum correntropy network: A trainable framework for adaptive filtering","name":"articletitle","label":"Article Title"},{"value":"Signal Processing","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.sigpro.2026.110723","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110723"}}