{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T09:58:35Z","timestamp":1770890315422,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T00:00:00Z","timestamp":1676419200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51875457"],"award-info":[{"award-number":["51875457"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022GY-050"],"award-info":[{"award-number":["2022GY-050"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022GY-028"],"award-info":[{"award-number":["2022GY-028"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022JQ-636"],"award-info":[{"award-number":["2022JQ-636"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021JQ-701"],"award-info":[{"award-number":["2021JQ-701"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2021JQ-714"],"award-info":[{"award-number":["2021JQ-714"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["20220129"],"award-info":[{"award-number":["20220129"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Research Project of Shaanxi Province","award":["51875457"],"award-info":[{"award-number":["51875457"]}]},{"name":"Key Research Project of Shaanxi Province","award":["2022GY-050"],"award-info":[{"award-number":["2022GY-050"]}]},{"name":"Key Research Project of Shaanxi Province","award":["2022GY-028"],"award-info":[{"award-number":["2022GY-028"]}]},{"name":"Key Research Project of Shaanxi Province","award":["2022JQ-636"],"award-info":[{"award-number":["2022JQ-636"]}]},{"name":"Key Research Project of Shaanxi Province","award":["2021JQ-701"],"award-info":[{"award-number":["2021JQ-701"]}]},{"name":"Key Research Project of Shaanxi Province","award":["2021JQ-714"],"award-info":[{"award-number":["2021JQ-714"]}]},{"name":"Key Research Project of Shaanxi Province","award":["20220129"],"award-info":[{"award-number":["20220129"]}]},{"name":"Natural Science Foundation of Shaanxi Province of China","award":["51875457"],"award-info":[{"award-number":["51875457"]}]},{"name":"Natural Science Foundation of Shaanxi Province of China","award":["2022GY-050"],"award-info":[{"award-number":["2022GY-050"]}]},{"name":"Natural Science Foundation of Shaanxi Province of China","award":["2022GY-028"],"award-info":[{"award-number":["2022GY-028"]}]},{"name":"Natural Science Foundation of Shaanxi Province of China","award":["2022JQ-636"],"award-info":[{"award-number":["2022JQ-636"]}]},{"name":"Natural Science Foundation of Shaanxi Province of China","award":["2021JQ-701"],"award-info":[{"award-number":["2021JQ-701"]}]},{"name":"Natural Science Foundation of Shaanxi Province of China","award":["2021JQ-714"],"award-info":[{"award-number":["2021JQ-714"]}]},{"name":"Natural Science Foundation of Shaanxi Province of China","award":["20220129"],"award-info":[{"award-number":["20220129"]}]},{"name":"Shaanxi Youth Talent Lifting Plan of Shaanxi Association for Science and Technology","award":["51875457"],"award-info":[{"award-number":["51875457"]}]},{"name":"Shaanxi Youth Talent Lifting Plan of Shaanxi Association for Science and Technology","award":["2022GY-050"],"award-info":[{"award-number":["2022GY-050"]}]},{"name":"Shaanxi Youth Talent Lifting Plan of Shaanxi Association for Science and Technology","award":["2022GY-028"],"award-info":[{"award-number":["2022GY-028"]}]},{"name":"Shaanxi Youth Talent Lifting Plan of Shaanxi Association for Science and Technology","award":["2022JQ-636"],"award-info":[{"award-number":["2022JQ-636"]}]},{"name":"Shaanxi Youth Talent Lifting Plan of Shaanxi Association for Science and Technology","award":["2021JQ-701"],"award-info":[{"award-number":["2021JQ-701"]}]},{"name":"Shaanxi Youth Talent Lifting Plan of Shaanxi Association for Science and Technology","award":["2021JQ-714"],"award-info":[{"award-number":["2021JQ-714"]}]},{"name":"Shaanxi Youth Talent Lifting Plan of Shaanxi Association for Science and Technology","award":["20220129"],"award-info":[{"award-number":["20220129"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Extreme learning machines (ELMs) have recently attracted significant attention due to their fast training speeds and good prediction effect. However, ELMs ignore the inherent distribution of the original samples, and they are prone to overfitting, which fails at achieving good generalization performance. In this paper, based on expectile penalty and correntropy, an asymmetric C-loss function (called AC-loss) is proposed, which is non-convex, bounded, and relatively insensitive to noise. Further, a novel extreme learning machine called L1 norm robust regularized extreme learning machine with asymmetric C-loss (L1-ACELM) is presented to handle the overfitting problem. The proposed algorithm benefits from L1 norm and replaces the square loss function with the AC-loss function. The L1-ACELM can generate a more compact network with fewer hidden nodes and reduce the impact of noise. To evaluate the effectiveness of the proposed algorithm on noisy datasets, different levels of noise are added in numerical experiments. The results for different types of artificial and benchmark datasets demonstrate that L1-ACELM achieves better generalization performance compared to other state-of-the-art algorithms, especially when noise exists in the datasets.<\/jats:p>","DOI":"10.3390\/axioms12020204","type":"journal-article","created":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T03:09:21Z","timestamp":1676430561000},"page":"204","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["L1-Norm Robust Regularized Extreme Learning Machine with Asymmetric C-Loss for Regression"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9400-236X","authenticated-orcid":false,"given":"Qing","family":"Wu","sequence":"first","affiliation":[{"name":"School of Automation, Xi\u2019an University of Posts and Telecommunications, Xi\u2019an 710121, China"},{"name":"Xi\u2019an Key Laboratory of Advanced Control and Intelligent Process, Xi\u2019an 710121, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Automation, Xi\u2019an University of Posts and Telecommunications, Xi\u2019an 710121, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu","family":"An","sequence":"additional","affiliation":[{"name":"School of Electronic Engineering, Xi\u2019an University of Posts and Telecommunications, Xi\u2019an 710121, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ke","family":"Li","sequence":"additional","affiliation":[{"name":"School of Automation, Xi\u2019an University of Posts and Telecommunications, Xi\u2019an 710121, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s10462-011-9208-z","article-title":"An optimizing BP neural network algorithm based on genetic algorithm","volume":"36","author":"Ding","year":"2011","journal-title":"Artif. Intell. Rev."},{"key":"ref_2","unstructured":"Huang, G.B., Zhu, Q.Y., and Siew, C.K. (2004, January 25\u201329). Extreme learning machine: A new learning scheme of feedforward neural networks. Proceedings of the 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No. 04CH37541), Budapest, Hungary."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.neucom.2005.12.126","article-title":"Extreme learning machine: Theory and applications","volume":"70","author":"Huang","year":"2006","journal-title":"Neurocomputing"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"112877","DOI":"10.1016\/j.eswa.2019.112877","article-title":"Outlier robust extreme machine learning for multi-target regression","volume":"140","author":"Silva","year":"2020","journal-title":"Expert Syst. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"106468","DOI":"10.1016\/j.knosys.2020.106468","article-title":"Bayesian robust multi-extreme learning machine","volume":"210","author":"Li","year":"2020","journal-title":"Knowl. -Based Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"108869","DOI":"10.1016\/j.petrol.2021.108869","article-title":"Extreme learning machine for multivariate reservoir characterization","volume":"205","author":"Liu","year":"2021","journal-title":"J. Pet. Sci. Eng."},{"key":"ref_7","first-page":"1148","article-title":"Challenging the empirical mean and empirical variance: A deviation study","volume":"48","author":"Catoni","year":"2012","journal-title":"Annales de l\u2019IHP Probabilit\u00e9s et Statistiques"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Deng, W., Zheng, Q., and Chen, L. (April, January 30). Regularized extreme learning machine. Proceedings of the 2009 IEEE Symposium on Computational Intelligence and Data Mining, Nashville, TN, USA.","DOI":"10.1109\/CIDM.2009.4938676"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/j.neucom.2008.01.005","article-title":"A fast pruned-extreme learning machine for classification problem","volume":"72","author":"Rong","year":"2008","journal-title":"Neurocomputing"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1109\/TNN.2009.2036259","article-title":"OP-ELM: Optimally pruned extreme learning machine","volume":"21","author":"Miche","year":"2009","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1109\/TCSVT.2016.2596158","article-title":"L1-norm distance linear discriminant analysis based on an effective iterative algorithm","volume":"28","author":"Ye","year":"2016","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1080\/02331934.2014.994627","article-title":"Robust L1-norm non-parallel proximal support vector machine","volume":"65","author":"Li","year":"2016","journal-title":"Optimization"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.neucom.2013.03.051","article-title":"1-Norm extreme learning machine for regression and multiclass classification using Newton method","volume":"128","author":"Balasundaram","year":"2014","journal-title":"Neurocomputing"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.1007\/s10115-021-01554-8","article-title":"Kernel-based regression via a novel robust loss function and iteratively reweighted least squares","volume":"63","author":"Dong","year":"2021","journal-title":"Knowl. Inf. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2881","DOI":"10.3233\/JIFS-191429","article-title":"Training robust support vector regression machines for more general noise","volume":"39","author":"Dong","year":"2020","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.csda.2016.11.010","article-title":"An SVM-like approach for expectile regression","volume":"109","author":"Farooq","year":"2017","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/j.future.2020.05.045","article-title":"Randomized nonlinear one-class support vector machines with bounded loss function to detect of outliers for large scale IoT data","volume":"112","author":"Razzak","year":"2020","journal-title":"Future Gener. Comput. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"12971","DOI":"10.1007\/s00521-020-04741-w","article-title":"Robust regularized extreme learning machine with asymmetric Huber loss function","volume":"32","author":"Gupta","year":"2020","journal-title":"Neural Comput. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.neucom.2018.05.100","article-title":"Correntropy-based robust extreme learning machine for classification","volume":"313","author":"Ren","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"70319","DOI":"10.1109\/ACCESS.2019.2919185","article-title":"LINEX support vector machine for large-scale classification","volume":"7","author":"Ma","year":"2019","journal-title":"IEEE Access."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.patcog.2013.07.017","article-title":"The C-loss function for pattern classification","volume":"47","author":"Singh","year":"2014","journal-title":"Pattern Recognit."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Zhou, R., Liu, X., Yu, M., and Huang, K. (2017). Properties of risk measures of generalized entropy in portfolio selection. Entropy, 19.","DOI":"10.3390\/e19120657"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s12859-020-03790-1","article-title":"Correntropy induced loss based sparse robust graph regularized extreme learning machine for cancer classification","volume":"21","author":"Ren","year":"2020","journal-title":"BMC Bioinform."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.ast.2019.04.023","article-title":"C-loss based extreme learning machine for estimating power of small-scale turbojet engine","volume":"89","author":"Zhao","year":"2019","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1109\/TSP.2019.2952057","article-title":"Robust matrix completion via maximum correntropy criterion and half-quadratic optimization","volume":"68","author":"He","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1543","DOI":"10.1007\/s11063-018-9890-9","article-title":"Robust extreme learning machines with different loss functions","volume":"49","author":"Ren","year":"2019","journal-title":"Neural Process. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.patcog.2018.07.011","article-title":"Correntropy-based robust multilayer extreme learning machines","volume":"84","author":"Chen","year":"2018","journal-title":"Pattern Recognit."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.neunet.2014.10.001","article-title":"Trends in extreme learning machines: A review","volume":"61","author":"Huang","year":"2015","journal-title":"Neural Netw."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1078","DOI":"10.1137\/17M114635X","article-title":"Inexact half-quadratic optimization for linear inverse problems","volume":"11","author":"Robini","year":"2018","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_30","unstructured":"Blake, C.L., Merz, C.J., and UCI Repository for Machine Learning Databases (2022, June 15). Department of Information and Computer Sciences, University of California, Irvine. Available online: http:\/\/www.ics.uci.edu\/~mlearn\/MLRepository.html."},{"key":"ref_31","first-page":"1","article-title":"Statistical comparisons of classifiers over multiple data sets","volume":"7","year":"2006","journal-title":"J. Mach. Learn. Res."},{"key":"ref_32","first-page":"152","article-title":"Should we really use post-hoc tests based on mean-ranks?","volume":"17","author":"Benavoli","year":"2016","journal-title":"J. Mach. Learn. Res."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/204\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:36:07Z","timestamp":1760121367000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/204"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,15]]},"references-count":32,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["axioms12020204"],"URL":"https:\/\/doi.org\/10.3390\/axioms12020204","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,15]]}}}