{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:27:33Z","timestamp":1787329653603,"version":"build-2736575974"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100007567","name":"City University of Hong Kong","doi-asserted-by":"publisher","award":["7006014"],"award-info":[{"award-number":["7006014"]}],"id":[{"id":"10.13039\/100007567","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["FRF-TP-22-105A1"],"award-info":[{"award-number":["FRF-TP-22-105A1"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["20223BCJ25017"],"award-info":[{"award-number":["20223BCJ25017"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee","doi-asserted-by":"publisher","award":["11300519"],"award-info":[{"award-number":["11300519"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee","doi-asserted-by":"publisher","award":["11300721"],"award-info":[{"award-number":["11300721"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee","doi-asserted-by":"publisher","award":["11311822"],"award-info":[{"award-number":["11311822"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008778","name":"University of Science and Technology Beijing","doi-asserted-by":"publisher","award":["12401332"],"award-info":[{"award-number":["12401332"]}],"id":[{"id":"10.13039\/501100008778","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"publisher","award":["12371297"],"award-info":[{"award-number":["12371297"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Journal on Mathematics of Data Science"],"published-print":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We consider kernel-based regularized learning with the help of the random projection technique to ease its computational burden. The random projection approaches we study here include the randomized sketching method and the Nystr\u00f6m approximation method. Current works under the least squares loss demonstrated an optimal learning rate under an appropriate source condition and capacity condition. However, beyond this simplest loss, it seems challenging to appropriately incorporate both conditions due to the unavailability of a closed-form solution. In this work, we consider a sufficiently general class of convex losses which include logistic loss, quantile loss, and hinge loss, for example, and establish the same optimal learning rate as in the least squares case under mild regularity assumptions. Our result also covers the unattainable case where the true function is not in the reproducing kernel Hilbert space. To incorporate the source condition, Young\u2019s inequality for operators is used, while to characterize the capacity, Rademacher complexity is adopted. We illustrate the performances of random projection with some numerical examples.<\/jats:p>","DOI":"10.1137\/23m1602954","type":"journal-article","created":{"date-parts":[[2025,2,6]],"date-time":"2025-02-06T09:52:02Z","timestamp":1738835522000},"page":"253-273","source":"Crossref","is-referenced-by-count":1,"title":["Kernel-Based Regularized Learning with Random Projections: Beyond Least Squares"],"prefix":"10.1137","volume":"7","author":[{"given":"Jiamin","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, 100083 Beijing, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Junzhuo","family":"Gao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, City University of Hong Kong, Hong Kong, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Heng","family":"Lian","sequence":"additional","affiliation":[{"name":"Department of Mathematics, City University of Hong Kong, Hong Kong, People\u2019s Republic of China, and Shenzhen Research Institute, City University of Hong Kong, 518000 Shenzhen, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,2,6]]},"reference":[{"key":"ref1","volume-title":"Advances in Neural Information Processing Systems","volume":"2015","author":"Alaoui A. 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