{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:10:44Z","timestamp":1760177444080,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,5,27]],"date-time":"2020-05-27T00:00:00Z","timestamp":1590537600000},"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":["11971466","11991021","11991020"],"award-info":[{"award-number":["11971466","11991021","11991020"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Key Research Program of Frontier Sciences, Chinese Academy of Sciences","award":["ZDBS-LY-7022"],"award-info":[{"award-number":["ZDBS-LY-7022"]}]},{"name":"the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy \tof Sciences and the Youth Innovation Promotion Association, Chinese Academy of Sciences"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Sparse dictionary learning (SDL) is a classic representation learning method and has been widely used in data analysis. Recently, the     \u2113 m    -norm (    m \u2265 3 , m \u2208 N    ) maximization has been proposed to solve SDL, which reshapes the problem to an optimization problem with orthogonality constraints. In this paper, we first propose an     \u2113 m    -norm maximization model for solving dual principal component pursuit (DPCP) based on the similarities between DPCP and SDL. Then, we propose a smooth unconstrained exact penalty model and show its equivalence with the     \u2113 m    -norm maximization model. Based on our penalty model, we develop an efficient first-order algorithm for solving our penalty model (PenNMF) and show its global convergence. Extensive experiments illustrate the high efficiency of PenNMF when compared with the other state-of-the-art algorithms on solving the     \u2113 m    -norm maximization with orthogonality constraints.<\/jats:p>","DOI":"10.3390\/s20113041","type":"journal-article","created":{"date-parts":[[2020,5,28]],"date-time":"2020-05-28T12:36:58Z","timestamp":1590669418000},"page":"3041","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Efficient Orthonormalization-Free Approach for Sparse Dictionary Learning and Dual Principal Component Pursuit"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4451-2019","authenticated-orcid":false,"given":"Xiaoyin","family":"Hu","sequence":"first","affiliation":[{"name":"Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5705-0805","authenticated-orcid":false,"given":"Xin","family":"Liu","sequence":"additional","affiliation":[{"name":"Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"},{"name":"State Key Laboratory of Scientific and Engineering Computing, Beijing 100190, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hansen, T.L., Badiu, M.A., Fleury, B.H., and Rao, B.D. 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