{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:23:47Z","timestamp":1760149427920,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:00:00Z","timestamp":1690156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12061071"],"award-info":[{"award-number":["12061071"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>For multi-class classification problems, a new kernel-free nonlinear classifier is presented, called the hard quadratic surface least squares regression (HQSLSR). It combines the benefits of the least squares loss function and quadratic kernel-free trick. The optimization problem of HQSLSR is convex and unconstrained, making it easy to solve. Further, to improve the generalization ability of HQSLSR, a softened version (SQSLSR) is proposed by introducing an \u03b5-dragging technique, which can enlarge the between-class distance. The optimization problem of SQSLSR is solved by designing an alteration iteration algorithm. The convergence, interpretability and computational complexity of our methods are addressed in a theoretical analysis. The visualization results on five artificial datasets demonstrate that the obtained regression function in each category has geometric diversity and the advantage of the \u03b5-dragging technique. Furthermore, experimental results on benchmark datasets show that our methods perform comparably to some state-of-the-art classifiers.<\/jats:p>","DOI":"10.3390\/e25071103","type":"journal-article","created":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T01:21:27Z","timestamp":1690248087000},"page":"1103","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Kernel-Free Quadratic Surface Regression for Multi-Class Classification"],"prefix":"10.3390","volume":"25","author":[{"given":"Changlin","family":"Wang","sequence":"first","affiliation":[{"name":"College of Mathematics and Systems Science, Xinjiang University, Urumuqi 830046, China"},{"name":"Institute of Mathematics and Physics, Xinjiang University, Urumuqi 830046, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0446-4278","authenticated-orcid":false,"given":"Zhixia","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Xinjiang University, Urumuqi 830046, China"},{"name":"Institute of Mathematics and Physics, Xinjiang University, Urumuqi 830046, China"}]},{"given":"Junyou","family":"Ye","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Xinjiang University, Urumuqi 830046, China"},{"name":"Institute of Mathematics and Physics, Xinjiang University, Urumuqi 830046, China"}]},{"given":"Xue","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Xinjiang University, Urumuqi 830046, China"},{"name":"Institute of Mathematics and Physics, Xinjiang University, Urumuqi 830046, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1255","DOI":"10.1080\/01621459.1994.10476866","article-title":"Flexible discriminant analysis by optimal scoring","volume":"89","author":"Hastie","year":"1993","journal-title":"J. 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