{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T23:54:22Z","timestamp":1778630062509,"version":"3.51.4"},"reference-count":24,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,6,29]],"date-time":"2019-06-29T00:00:00Z","timestamp":1561766400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Hubei Province in China","award":["2017CFB523"],"award-info":[{"award-number":["2017CFB523"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11671307,11571078"],"award-info":[{"award-number":["11671307,11571078"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the framework of statistical learning, we study the online gradient descent algorithm generated by the correntropy-induced losses in Reproducing kernel Hilbert spaces (RKHS). As a generalized correlation measurement, correntropy has been widely applied in practice, owing to its prominent merits on robustness. Although the online gradient descent method is an efficient way to deal with the maximum correntropy criterion (MCC) in non-parameter estimation, there has been no consistency in analysis or rigorous error bounds. We provide a theoretical understanding of the online algorithm for MCC, and show that, with a suitable chosen scaling parameter, its convergence rate can be min\u2013max optimal (up to a logarithmic factor) in the regression analysis. Our results show that the scaling parameter plays an essential role in both robustness and consistency.<\/jats:p>","DOI":"10.3390\/e21070644","type":"journal-article","created":{"date-parts":[[2019,7,1]],"date-time":"2019-07-01T03:23:59Z","timestamp":1561951439000},"page":"644","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Online Gradient Descent for Kernel-Based Maximum Correntropy Criterion"],"prefix":"10.3390","volume":"21","author":[{"given":"Baobin","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3770-6309","authenticated-orcid":false,"given":"Ting","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2187","DOI":"10.1109\/TSP.2006.872524","article-title":"Generalized correlation function: Definition, properties, and application to blind equalization","volume":"54","author":"Santamaria","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_2","first-page":"993","article-title":"Learning with the Maximum Correntropy Criterion Induced Losses for Regression","volume":"16","author":"Feng","year":"2015","journal-title":"J. Mach. Learn. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1561","DOI":"10.1109\/TPAMI.2010.220","article-title":"Maximum Correntropy Criterion for Robust Face Recognition","volume":"33","author":"He","year":"2011","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_4","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."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Principe, J.C. (2010). Renyi\u2019s Entropy and Kernel Perspectives. Information Theoretic Learning, Springer.","DOI":"10.1007\/978-1-4419-1570-2"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2074","DOI":"10.1162\/NECO_a_00155","article-title":"A regularized correntropy framework for robust pattern recognition","volume":"23","author":"He","year":"2011","journal-title":"Neural Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1657","DOI":"10.1109\/TPWRS.2009.2030291","article-title":"Entropy and correntropy against minimum square error in offline and online three-day ahead wind power forecasting","volume":"24","author":"Bessa","year":"2009","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1109\/TIP.2010.2103949","article-title":"Robust Principal Component Analysis Based on Maximum Correntropy Criterion","volume":"20","author":"He","year":"2011","journal-title":"IEEE Trans. Image Process."},{"key":"ref_9","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":"ref_10","doi-asserted-by":"crossref","first-page":"7149","DOI":"10.3390\/e17107149","article-title":"Robust Hammerstein Adaptive Filtering under Maximum Correntropy Criterion","volume":"17","author":"Wu","year":"2015","journal-title":"Entropy"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Liu, W., Pokharel, P.P., and Principe, J.C. (2006, January 6\u20138). Error Entropy, Correntropy and M-Estimation. Proceedings of the 16th IEEE Signal Processing Society Workshop on Machine Learning for Signal Processing, Arlington, VA, USA.","DOI":"10.1109\/MLSP.2006.275544"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1109\/LSP.2013.2283425","article-title":"Invexity of the minimum error entropy criterion","volume":"20","author":"Syed","year":"2013","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1007\/s11590-013-0626-5","article-title":"On the optimization properties of the correntropic loss function in data analysis","volume":"8","author":"Syed","year":"2014","journal-title":"Optim. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Marques de S\u00e1, J.P., Silva, L.M.A., Santos, J.M.F., and Alexandre, L.A. (2013). Minimum Error Entropy Classification, Studies in Computational Intelligence; Springer.","DOI":"10.1007\/978-3-642-29029-9"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Cucker, F., and Zhou, D.X. (2007). Learning Theory: An Approximation Theory Viewpoint, Cambridge University Press.","DOI":"10.1017\/CBO9780511618796"},{"key":"ref_16","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_17","doi-asserted-by":"crossref","unstructured":"Guo, Z.C., Hu, T., and Shi, L. (2018). Gradient descent for robust kernel-based regression. Inverse Prob., 34.","DOI":"10.1088\/1361-6420\/aabe55"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s00365-006-0659-y","article-title":"Learning theory estimates via integral operators and their approximations","volume":"26","author":"Smale","year":"2007","journal-title":"Constr. Approx."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Lu, S., and Pereverzev, S.V. (2013). Regularization Theory for Ill-Posed Problems: Selected Topics, Walter de Gruyter.","DOI":"10.1515\/9783110286496"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s10208-006-0196-8","article-title":"Optimal rates for the regularized least-squares algorithm","volume":"7","author":"Caponnetto","year":"2007","journal-title":"Found. Comput. Math."},{"key":"ref_21","unstructured":"Steinwart, I., and Christmann, A. (2008). Support Vector Machines, Springer."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.jco.2006.07.001","article-title":"On regularization algorithms in learning theory","volume":"23","author":"Bauer","year":"2007","journal-title":"J. Complexity"},{"key":"ref_23","unstructured":"Feng, Y.L., Fan, J., and Suykens, J.A. (2017). A statistical learning approach to modal regression. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1007\/s10208-006-0237-y","article-title":"Online gradient descent learning algorithms","volume":"8","author":"Ying","year":"2008","journal-title":"Found. Comput. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/7\/644\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:02:22Z","timestamp":1760187742000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/7\/644"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,29]]},"references-count":24,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2019,7]]}},"alternative-id":["e21070644"],"URL":"https:\/\/doi.org\/10.3390\/e21070644","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,29]]}}}