{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T04:31:36Z","timestamp":1783744296051,"version":"3.55.0"},"reference-count":54,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T00:00:00Z","timestamp":1682553600000},"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":["62062067"],"award-info":[{"award-number":["62062067"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Recently, there has been a rapid increase in deep classification tasks, such as image recognition and target detection. As one of the most crucial components in Convolutional Neural Network (CNN) architectures, softmax arguably encourages CNN to achieve better performance in image recognition. Under this scheme, we present a conceptually intuitive learning objection function: Orthogonal-Softmax. The primary property of the loss function is to use a linear approximation model that is designed by Gram\u2013Schmidt orthogonalization. Firstly, compared with the traditional softmax and Taylor-Softmax, Orthogonal-Softmax has a stronger relationship through orthogonal polynomials expansion. Secondly, a new loss function is advanced to acquire highly discriminative features for classification tasks. At last, we present a linear softmax loss to further promote the intra-class compactness and inter-class discrepancy simultaneously. The results of the widespread experimental discussion on four benchmark datasets manifest the validity of the presented method. Besides, we want to explore the non-ground truth samples in the future.<\/jats:p>","DOI":"10.3390\/e25050727","type":"journal-article","created":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T02:27:56Z","timestamp":1682648876000},"page":"727","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Deep Classification with Linearity-Enhanced Logits to Softmax Function"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2687-7233","authenticated-orcid":false,"given":"Hao","family":"Shao","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Yunnan Unverisity, Kunming 650504, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1927-8753","authenticated-orcid":false,"given":"Shunfang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Information Science and Engineering, Yunnan Unverisity, Kunming 650504, China"},{"name":"The Key Lab of Intelligent Systems and Computing of Yunnan Province, Yunnan University, Kunming 650504, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.neucom.2020.10.081","article-title":"Deep face recognition: A survey","volume":"429","author":"Wang","year":"2021","journal-title":"Neurocomputing"},{"key":"ref_2","unstructured":"Krizhevsky, A., Sutskever, I., and Hinton, G.E. 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