{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T04:26:21Z","timestamp":1773894381130,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T00:00:00Z","timestamp":1696896000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100011447","name":"Henan Center for Outstanding Overseas Scientists","doi-asserted-by":"publisher","award":["GZS2020012"],"award-info":[{"award-number":["GZS2020012"]}],"id":[{"id":"10.13039\/501100011447","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011447","name":"Henan Center for Outstanding Overseas Scientists","doi-asserted-by":"publisher","award":["K2022TD001"],"award-info":[{"award-number":["K2022TD001"]}],"id":[{"id":"10.13039\/501100011447","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011447","name":"Henan Center for Outstanding Overseas Scientists","doi-asserted-by":"publisher","award":["232102211020"],"award-info":[{"award-number":["232102211020"]}],"id":[{"id":"10.13039\/501100011447","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011447","name":"Henan Center for Outstanding Overseas Scientists","doi-asserted-by":"publisher","award":["222102210016"],"award-info":[{"award-number":["222102210016"]}],"id":[{"id":"10.13039\/501100011447","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Machine Intelligence and High-Dimensional Data Analysis","award":["GZS2020012"],"award-info":[{"award-number":["GZS2020012"]}]},{"name":"Machine Intelligence and High-Dimensional Data Analysis","award":["K2022TD001"],"award-info":[{"award-number":["K2022TD001"]}]},{"name":"Machine Intelligence and High-Dimensional Data Analysis","award":["232102211020"],"award-info":[{"award-number":["232102211020"]}]},{"name":"Machine Intelligence and High-Dimensional Data Analysis","award":["222102210016"],"award-info":[{"award-number":["222102210016"]}]},{"DOI":"10.13039\/501100011447","name":"Key Technologies R&amp;D Program of Henan","doi-asserted-by":"publisher","award":["GZS2020012"],"award-info":[{"award-number":["GZS2020012"]}],"id":[{"id":"10.13039\/501100011447","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011447","name":"Key Technologies R&amp;D Program of Henan","doi-asserted-by":"publisher","award":["K2022TD001"],"award-info":[{"award-number":["K2022TD001"]}],"id":[{"id":"10.13039\/501100011447","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011447","name":"Key Technologies R&amp;D Program of Henan","doi-asserted-by":"publisher","award":["232102211020"],"award-info":[{"award-number":["232102211020"]}],"id":[{"id":"10.13039\/501100011447","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011447","name":"Key Technologies R&amp;D Program of Henan","doi-asserted-by":"publisher","award":["222102210016"],"award-info":[{"award-number":["222102210016"]}],"id":[{"id":"10.13039\/501100011447","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>To improve the accuracy of color image completion with missing entries, we present a recovery method based on generalized higher-order scalars. We extend the traditional second-order matrix model to a more comprehensive higher-order matrix equivalent, called the \u201ct-matrix\u201d model, which incorporates a pixel neighborhood expansion strategy to characterize the local pixel constraints. This \u201ct-matrix\u201d model is then used to extend some commonly used matrix and tensor completion algorithms to their higher-order versions. We perform extensive experiments on various algorithms using simulated data and publicly available images. The results show that our generalized matrix completion model and the corresponding algorithm compare favorably with their lower-order tensor and conventional matrix counterparts.<\/jats:p>","DOI":"10.3390\/axioms12100954","type":"journal-article","created":{"date-parts":[[2023,10,11]],"date-time":"2023-10-11T02:11:01Z","timestamp":1696990261000},"page":"954","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":66,"title":["Color Image Recovery Using Generalized Matrix Completion over Higher-Order Finite Dimensional Algebra"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4613-7200","authenticated-orcid":false,"given":"Liang","family":"Liao","sequence":"first","affiliation":[{"name":"School of Electronics and Information, Zhongyuan University of Technology, Zhengzhou 451191, China"}]},{"given":"Zhuang","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Electronics and Information, Zhongyuan University of Technology, Zhengzhou 451191, China"}]},{"given":"Qi","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Electronics and Information, Zhongyuan University of Technology, Zhengzhou 451191, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9070-6653","authenticated-orcid":false,"given":"Yan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electronics and Information, Zhongyuan University of Technology, Zhengzhou 451191, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5986-8419","authenticated-orcid":false,"given":"Fajun","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Electronics and Information, Zhongyuan University of Technology, Zhengzhou 451191, China"}]},{"given":"Qifeng","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Electronics and Information, Zhongyuan University of Technology, Zhengzhou 451191, China"}]},{"given":"Stephen John","family":"Maybank","sequence":"additional","affiliation":[{"name":"Birkbeck College, University of London, London WC1E 7HY, UK"}]},{"given":"Zhoufeng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Electronics and Information, Zhongyuan University of Technology, Zhengzhou 451191, China"}]},{"given":"Chunlei","family":"Li","sequence":"additional","affiliation":[{"name":"School of Electronics and Information, Zhongyuan University of Technology, Zhengzhou 451191, China"}]},{"given":"Lun","family":"Li","sequence":"additional","affiliation":[{"name":"School of Information Engineering, Zhengzhou University, Zhengzhou 450001, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1038\/s42256-022-00575-4","article-title":"Data-driven discovery of intrinsic dynamics","volume":"4","author":"Floryan","year":"2022","journal-title":"Nat. Mach. Intell."},{"key":"ref_2","first-page":"3411","article-title":"Low dimensional trajectory hypothesis is true: DNNs can be trained in tiny subspaces","volume":"45","author":"Li","year":"2022","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_3","first-page":"1203","article-title":"Nonparametric regression on low-dimensional manifolds using deep ReLU networks: Function approximation and statistical recovery","volume":"11","author":"Chen","year":"2022","journal-title":"Inf. Inference J. IMA"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1109\/TKDE.2021.3087517","article-title":"A novel classifier ensemble method based on subspace enhancement for high-dimensional data classification","volume":"35","author":"Xu","year":"2021","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1016\/j.ins.2022.05.091","article-title":"Low-rank tensor approximation with local structure for multi-view intrinsic subspace clustering","volume":"606","author":"Fu","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1109\/TCYB.2022.3175771","article-title":"Learning tensor low-rank representation for hyperspectral anomaly detection","volume":"53","author":"Wang","year":"2022","journal-title":"IEEE Trans. Cybern."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"140302","DOI":"10.1007\/s11432-022-3609-4","article-title":"A survey on hyperspectral image restoration: From the view of low-rank tensor approximation","volume":"66","author":"Liu","year":"2023","journal-title":"Sci. China Inf. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1109\/TPAMI.2012.39","article-title":"Tensor Completion for Estimating Missing Values in Visual Data","volume":"35","author":"Liu","year":"2012","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhang, Z., Ely, G., Aeron, S., Hao, N., and Kilmer, M. (2014, January 23\u201328). Novel methods for multilinear data completion and de-noising based on tensor-SVD. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Columbus, OH, USA.","DOI":"10.1109\/CVPR.2014.485"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1016\/j.laa.2010.09.020","article-title":"Factorization strategies for third-order tensors","volume":"435","author":"Kilmer","year":"2011","journal-title":"Linear Algebra Its Appl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Lu, C., Feng, J., Lin, Z., and Yan, S. (2018). Exact low tubal rank tensor recovery from Gaussian measurements. arXiv.","DOI":"10.24963\/ijcai.2018\/347"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Xue, S., Qiu, W., Liu, F., and Jin, X. (2018, January 20\u201324). Low-rank tensor completion by truncated nuclear norm regularization. Proceedings of the 2018 24th IEEE International Conference on Pattern Recognition (ICPR), Beijing, China.","DOI":"10.1109\/ICPR.2018.8546008"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zeng, H., Xue, J., Luong, H.Q., and Philips, W. (IEEE Trans. Multimed., 2022). Multimodal core tensor factorization and its applications to low-rank tensor completion, IEEE Trans. Multimed., early access.","DOI":"10.1109\/TMM.2022.3216746"},{"key":"ref_14","first-page":"27008","article-title":"Tensor wheel decomposition and its tensor completion application","volume":"35","author":"Wu","year":"2022","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1137\/21M1429539","article-title":"Robust tensor completion: Equivalent surrogates, error bounds, and algorithms","volume":"15","author":"Zhao","year":"2022","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_16","first-page":"4038","article-title":"A New Automatic Hyperparameter Recommendation Approach Under Low-Rank Tensor Completion e Framework","volume":"45","author":"Deng","year":"2022","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3604649","article-title":"Tensor completion with provable consistency and fairness guarantees for recommender systems","volume":"1","author":"Nguyen","year":"2023","journal-title":"ACM Trans. Recomm. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Hui, B., Yan, D., Chen, H., and Ku, W.-S. (2022, January 9\u201312). Time-sensitive POI Recommendation by Tensor Completion with Side Information. Proceedings of the 2022 IEEE 38th International Conference on Data Engineering (ICDE), Kuala Lumpur, Malaysia.","DOI":"10.1109\/ICDE53745.2022.00020"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3278607","article-title":"Tensor completion algorithms in big data analytics","volume":"13","author":"Song","year":"2019","journal-title":"ACM Trans. Knowl. Discov. Data"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"20203","DOI":"10.1109\/JIOT.2022.3171780","article-title":"A multi-attention tensor completion network for spatiotemporal traffic data imputation","volume":"9","author":"Wu","year":"2022","journal-title":"IEEE Internet Things J."},{"key":"ref_21","first-page":"21782","article-title":"Beyond the signs: Nonparametric tensor completion via sign series","volume":"34","author":"Lee","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1007\/s10851-020-00946-9","article-title":"Generalized visual information analysis via tensorial algebra","volume":"62","author":"Liao","year":"2020","journal-title":"J. Math. Imaging Vis."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/s40314-022-01811-8","article-title":"T-product tensors\u2014Part II: Tail bounds for sums of random T-product tensors","volume":"41","author":"Chang","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1007\/s10589-022-00439-y","article-title":"T-product factorization based method for matrix and tensor completion problems","volume":"84","author":"Yu","year":"2022","journal-title":"Comput. Optim. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1137\/110837711","article-title":"Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging","volume":"34","author":"Kilmer","year":"2021","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Lim, L.-H. (2013). Tensors and Hypermatrices, Handbook of Linear Algebra (Leslie Hogben, Ed.), CRC Press.","DOI":"10.1201\/b16113-19"},{"key":"ref_27","first-page":"1","article-title":"Robust principal component analysis?","volume":"58","author":"Li","year":"2021","journal-title":"J. ACM (JACM)"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1145\/2184319.2184343","article-title":"Exact matrix completion via convex optimization","volume":"55","author":"Recht","year":"2012","journal-title":"Commun. ACM"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Lin, Z., Li, H., and Fang, C. (2022). Alternating Direction Method of Multipliers for Machine Learning, Springer.","DOI":"10.1007\/978-981-16-9840-8"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40305-021-00368-3","article-title":"A survey on some recent developments of alternating direction method of multipliers","volume":"10","author":"Han","year":"2022","journal-title":"J. Oper. Res. Soc. China"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1109\/TPAMI.2019.2891760","article-title":"Tensor robust principal component analysis with a new tensor nuclear norm","volume":"42","author":"Lu","year":"2019","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_32","unstructured":"Liao, L., and Maybank, S.J. (2020). General data analytics with applications to visual information analysis: A provable backward-compatible semisimple paradigm over t-algebra. arXiv."},{"key":"ref_33","unstructured":"Liao, L., Lin, S., Li, L., Zhang, X., Zhao, S., Wang, Y., Wang, X., Gao, Q., and Wang, J. (2022). Approximation of Images via Generalized Higher Order Singular Value Decomposition over Finite-Dimensional Commutative Semisimple Algebra. arXiv."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/954\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:04:14Z","timestamp":1760130254000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/954"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,10]]},"references-count":33,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["axioms12100954"],"URL":"https:\/\/doi.org\/10.3390\/axioms12100954","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,10]]}}}