{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:49:05Z","timestamp":1791323345619,"version":"4.3.1"},"reference-count":36,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T00:00:00Z","timestamp":1520294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper studies the matrix completion problems when the entries are contaminated by non-Gaussian noise or outliers. The proposed approach employs a nonconvex loss function induced by the maximum correntropy criterion. With the help of this loss function, we develop a rank constrained, as well as a nuclear norm regularized model, which is resistant to non-Gaussian noise and outliers. However, its non-convexity also leads to certain difficulties. To tackle this problem, we use the simple iterative soft and hard thresholding strategies. We show that when extending to the general affine rank minimization problems, under proper conditions, certain recoverability results can be obtained for the proposed algorithms. Numerical experiments indicate the improved performance of our proposed approach.<\/jats:p>","DOI":"10.3390\/e20030171","type":"journal-article","created":{"date-parts":[[2018,3,6]],"date-time":"2018-03-06T12:16:27Z","timestamp":1520338587000},"page":"171","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Correntropy Based Matrix Completion"],"prefix":"10.3390","volume":"20","author":[{"given":"Yuning","family":"Yang","sequence":"first","affiliation":[{"name":"College of Mathematics and Information Science, Guangxi University, Nanning 530004, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yunlong","family":"Feng","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, The State University of New York at Albany, Albany, NY 12222, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Johan","family":"Suykens","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, ESAT-STADIUS, KU Leuven, Kasteelpark Arenberg 10, Leuven B-3001, Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,6]]},"reference":[{"key":"ref_1","unstructured":"Srebro, N., and Jaakkola, T. (2003, January 11\u201312). Weighted low-rank approximations. Proceedings of the 20th International Conference on Machine Learning, Copenhagen, Denmark."},{"key":"ref_2","unstructured":"Prize Website, N. (2018, March 02). Available online: http:\/\/www.netflixprize.com."},{"key":"ref_3","unstructured":"Komodakis, N. (2006, January 17\u201322). Image completion using global optimization. Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York, NY, USA."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bertalmio, M., Sapiro, G., Caselles, V., and Ballester, C. (2000, January 23\u201328). Image inpainting. Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, New Orleans, LA, USA.","DOI":"10.1145\/344779.344972"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ji, H., Liu, C., Shen, Z., and Xu, Y. (2010, January 13\u201318). Robust video denoising using low rank matrix completion. Proceedings of the 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), San Francisco, CA, USA.","DOI":"10.1109\/CVPR.2010.5539849"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1109\/JPROC.2009.2035722","article-title":"Matrix completion with noise","volume":"98","author":"Plan","year":"2010","journal-title":"Proc. IEEE"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1007\/s10208-009-9045-5","article-title":"Exact matrix completion via convex optimization","volume":"9","author":"Recht","year":"2009","journal-title":"Found. Comput. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1548","DOI":"10.1109\/TIT.2011.2104999","article-title":"Recovering low-rank matrices from few coefficients in any basis","volume":"57","author":"Gross","year":"2011","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_9","first-page":"2057","article-title":"Matrix completion from noisy entries","volume":"99","author":"Keshavan","year":"2010","journal-title":"J. Mach. Learn. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2302","DOI":"10.1214\/11-AOS894","article-title":"Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion","volume":"39","author":"Koltchinskii","year":"2011","journal-title":"Ann. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1109\/LSP.2011.2151856","article-title":"Tensor versus matrix completion: A comparison with application to spectral data","volume":"18","author":"Signoretto","year":"2011","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2117","DOI":"10.1109\/TPAMI.2012.271","article-title":"Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization","volume":"35","author":"Hu","year":"2013","journal-title":"IEEE Trans. Pattern Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1956","DOI":"10.1137\/080738970","article-title":"A singular value thresholding algorithm for matrix completion","volume":"20","author":"Cai","year":"2010","journal-title":"SIAM J. Optim."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s10208-011-9084-6","article-title":"Convergence of fixed-point continuation algorithms for matrix rank minimization","volume":"11","author":"Goldfarb","year":"2011","journal-title":"Found. Comput. Math."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ji, S., and Ye, J. (2009, January 14\u201318). An accelerated gradient method for trace norm minimization. Proceedings of the 26th Annual International Conference on Machine Learning, Montreal, QC, Canada.","DOI":"10.1145\/1553374.1553434"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s10107-009-0306-5","article-title":"Fixed point and Bregman iterative methods for matrix rank minimization","volume":"128","author":"Ma","year":"2011","journal-title":"Math. Program."},{"key":"ref_17","first-page":"11","article-title":"Robust principal component analysis?","volume":"58","author":"Li","year":"2011","journal-title":"J. ACM (JACM)"},{"key":"ref_18","unstructured":"Hastie, T. (2018, February 21). Matrix Completion and Large-Scale SVD Computations. Available online: http:\/\/www.stanford.edu\/~hastie\/TALKS\/SVD_hastie.pdf."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Nie, F., Wang, H., Cai, X., Huang, H., and Ding, C. (2012, January 10\u201313). Robust matrix completion via joint Schatten p-norm andlp-norm minimization. Proceedings of the 2012 IEEE 12th International Conference on Data Mining (ICDM), Brussels, Belgium.","DOI":"10.1109\/ICDM.2012.160"},{"key":"ref_20","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_21","doi-asserted-by":"crossref","unstructured":"Pr\u00edncipe, J.C. (2010). Information Theoretic Learning: Renyi\u2019s Entropy and Kernel Perspectives, Springer Science & Business Media.","DOI":"10.1007\/978-1-4419-1570-2"},{"key":"ref_22","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_23","unstructured":"Jain, P., Meka, R., and Dhillon, I.S. (2010, January 6\u201311). Guaranteed Rank Minimization via Singular Value Projection. Proceedings of the Advances in Neural Information Processing Systems, Hyatt Regency, VAN, Canada."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Beck, A., and Teboulle, M.A. (2011). A linearly convergent algorithm for solving a class of nonconvex\/affine feasibility problems. Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer.","DOI":"10.1007\/978-1-4419-9569-8_3"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1214\/10-AOS860","article-title":"Estimation of high-dimensional low-rank matrices","volume":"39","author":"Rohde","year":"2011","journal-title":"Ann. Stat."},{"key":"ref_26","unstructured":"Chen, Y., Xu, H., Caramanis, C., and Sanghavi, S. (arXiv, 2011). Robust Matrix Completion with Corrupted Columns, arXiv."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Huber, P.J. (2011). Robust Statistics, Springer.","DOI":"10.1007\/978-3-642-04898-2_594"},{"key":"ref_28","unstructured":"Warmuth, M.K. (2018, February 21). From Relative Entropies to Bregman Divergences to the Design of Convex and Tempered Non-Convex Losses. Available online: http:\/\/classes.soe.ucsc.edu\/cmps290c\/Spring13\/lect\/9\/holycow.pdf."},{"key":"ref_29","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_30","doi-asserted-by":"crossref","first-page":"3376","DOI":"10.1109\/TSP.2016.2539127","article-title":"Generalized correntropy for robust adaptive filtering","volume":"64","author":"Chen","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.automatica.2016.10.004","article-title":"Maximum correntropy Kalman filter","volume":"76","author":"Chen","year":"2017","journal-title":"Automatica"},{"key":"ref_32","unstructured":"Feng, Y., Fan, J., and Suykens, J. (arXiv, 2017). A Statistical Learning Approach to Modal Regression, arXiv."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1137\/070697835","article-title":"Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization","volume":"52","author":"Recht","year":"2010","journal-title":"SIAM Rev."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s10107-013-0701-9","article-title":"Proximal alternating linearized minimization for nonconvex and nonsmooth problems","volume":"146","author":"Bolte","year":"2014","journal-title":"Math. Program."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1459","DOI":"10.1109\/TIP.2004.836169","article-title":"Statistical modeling of complex backgrounds for foreground object detection","volume":"13","author":"Li","year":"2004","journal-title":"IEEE Trans. Image Process."},{"key":"ref_36","unstructured":"Wright, J., Ganesh, A., Rao, S., Peng, Y., and Ma, Y. (2009, January 7\u20138). Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization. Proceedings of the Advances in Neural Information Processing Systems, Vancouver, BC, Canada."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/3\/171\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:57:39Z","timestamp":1760194659000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/3\/171"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,6]]},"references-count":36,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,3]]}},"alternative-id":["e20030171"],"URL":"https:\/\/doi.org\/10.3390\/e20030171","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,3,6]]}}}