{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,9]],"date-time":"2025-11-09T07:44:14Z","timestamp":1762674254449,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,6,28]],"date-time":"2018-06-28T00:00:00Z","timestamp":1530144000000},"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":["61702262, 61703209, 91420201, 61472187"],"award-info":[{"award-number":["61702262, 61703209, 91420201, 61472187"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Many tasks in computer vision suffer from missing values in tensor data, i.e., multi-way data array. The recently proposed tensor tubal nuclear norm (TNN) has shown superiority in imputing missing values in 3D visual data, like color images and videos. However, by interpreting in a circulant way, TNN only exploits tube (often carrying temporal\/channel information) redundancy in a circulant way while preserving the row and column (often carrying spatial information) relationship. In this paper, a new tensor norm named the triple tubal nuclear norm (TriTNN) is proposed to simultaneously exploit tube, row and column redundancy in a circulant way by using a weighted sum of three TNNs. Thus, more spatial-temporal information can be mined. Further, a TriTNN-based tensor completion model with an ADMM solver is developed. Experiments on color images, videos and LiDAR datasets show the superiority of the proposed TriTNN against state-of-the-art nuclear norm-based tensor norms.<\/jats:p>","DOI":"10.3390\/a11070094","type":"journal-article","created":{"date-parts":[[2018,6,28]],"date-time":"2018-06-28T10:53:33Z","timestamp":1530183213000},"page":"94","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Tensor Completion Based on Triple Tubal Nuclear Norm"],"prefix":"10.3390","volume":"11","author":[{"given":"Dongxu","family":"Wei","sequence":"first","affiliation":[{"name":"School of Physics and Electronic Electrical Engineering, Huaiyin Normal University, Huaian 223300, China"},{"name":"School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"},{"name":"Jiangsu Province Key Construction Laboratory of Modern Measurement Technology and Intelligent System, Huaian 223300, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0412-2120","authenticated-orcid":false,"given":"Andong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}]},{"given":"Xiaoqin","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Physics and Electronic Electrical Engineering, Huaiyin Normal University, Huaian 223300, China"}]},{"given":"Boyu","family":"Wang","sequence":"additional","affiliation":[{"name":"Jiangsu Yaoshi Software Technology Co., Ltd., Nanjing 211103, China"}]},{"given":"Bo","family":"Wang","sequence":"additional","affiliation":[{"name":"Jiangsu Shuoshi Welding Technology Co., Ltd., Nanjing 211103, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,28]]},"reference":[{"key":"ref_1","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":"2013","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_2","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":"2013","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wang, A., and Jin, Z. (2017, January 18\u201321). Near-optimal Noisy Low-tubal-rank Tensor Completion via Singular Tube Thresholding. Proceedings of the IEEE International Conference on Data Mining Workshop (ICDMW), New Orleans, LA, USA.","DOI":"10.1109\/ICDMW.2017.78"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s10994-013-5366-3","article-title":"Learning with tensors: a framework based on convex optimization and spectral regularization","volume":"94","author":"Signoretto","year":"2014","journal-title":"Mach. Learn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1109\/MSP.2013.2297439","article-title":"Tensor decompositions for signal processing applications: From two-way to multiway component analysis","volume":"32","author":"Cichocki","year":"2015","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Lu, C., Feng, J., Chen, Y., Liu, W., Lin, Z., and Yan, S. (2016, January 27\u201330). Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Las Vegas, NV, USA.","DOI":"10.1109\/CVPR.2016.567"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"11559","DOI":"10.1109\/ACCESS.2018.2811396","article-title":"Tensor Completion Using Spectral (k, p) -Support Norm","volume":"6","author":"Wei","year":"2018","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1109\/LSP.2013.2245416","article-title":"An Efficient Matrix Factorization Method for Tensor Completion","volume":"20","author":"Liu","year":"2013","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Song, X., and Lu, H. (2017, January 4\u20139). Multilinear Regression for Embedded Feature Selection with Application to fMRI Analysis. Proceedings of the AAAI Conference on Artificial Intelligence, San Francisco, CA, USA.","DOI":"10.1609\/aaai.v31i1.10871"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.trc.2012.12.007","article-title":"A tensor-based method for missing traffic data completion","volume":"28","author":"Tan","year":"2013","journal-title":"Transp. Res. Part C"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1751","DOI":"10.1109\/TPAMI.2015.2392756","article-title":"Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination","volume":"37","author":"Zhao","year":"2015","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1561\/2200000059","article-title":"Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions","volume":"9","author":"Cichocki","year":"2016","journal-title":"Found. Trends Mach. Learn."},{"key":"ref_13","unstructured":"Harshman, R.A. (1970). Foundations of the PARAFAC Procedure: Models and Conditions for an \u201cExplanatory\u201d Multi-Modal Factor Analysis, University of California."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1137\/07070111X","article-title":"Tensor decompositions and applications","volume":"51","author":"Kolda","year":"2009","journal-title":"SIAM Rev."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/BF02289464","article-title":"Some mathematical notes on three-mode factor analysis","volume":"31","author":"Tucker","year":"1966","journal-title":"Psychometrika"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2961","DOI":"10.1109\/TNNLS.2016.2611525","article-title":"The twist tensor nuclear norm for video completion","volume":"28","author":"Hu","year":"2017","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6753","DOI":"10.1109\/TIT.2017.2724549","article-title":"Incoherent Tensor Norms and Their Applications in Higher Order Tensor Completion","volume":"63","author":"Yuan","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_18","unstructured":"Song, Q., Ge, H., Caverlee, J., and Hu, X. (arXiv, 2017). Tensor Completion Algorithms in Big Data Analytics, arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2053","DOI":"10.1109\/TIT.2010.2044061","article-title":"The power of convex relaxation: near-optimal matrix completion","volume":"56","author":"Tao","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_20","first-page":"45","article-title":"Most Tensor Problems Are NP-Hard","volume":"60","author":"Hillar","year":"2009","journal-title":"J. ACM"},{"key":"ref_21","unstructured":"Tomioka, R., and Suzuki, T. (2013, January 5\u201310). Convex tensor decomposition via structured schatten norm regularization. Proceedings of the Advances in Neural Information Processing Systems, Lake Tahoe, NV, USA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1678","DOI":"10.1109\/TIP.2014.2305840","article-title":"Tensor-Based Formulation and Nuclear Norm Regularization for Multienergy Computed Tomography","volume":"23","author":"Semerci","year":"2014","journal-title":"IEEE Trans. Image Process."},{"key":"ref_23","unstructured":"Mu, C., Huang, B., Wright, J., and Goldfarb, D. (2014, January 21\u201326). Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery. Proceedings of the International Conference on Machine Learning, Beijing, China."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Zhao, Q., Meng, D., Kong, X., Xie, Q., Cao, W., Wang, Y., and Xu, Z. (2015, January 7\u201313). A Novel Sparsity Measure for Tensor Recovery. Proceedings of the IEEE International Conference on Computer Vision, Santiago, Chile.","DOI":"10.1109\/ICCV.2015.39"},{"key":"ref_25","unstructured":"Tomioka, R., Hayashi, K., and Kashima, H. (arXiv, 2010). Estimation of low-rank tensors via convex optimization, arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1109\/TIT.2016.2633413","article-title":"Sensing tensors with Gaussian filters","volume":"63","author":"Chretien","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Ghadermarzy, N., Plan, Y., and Y\u0131lmaz, \u00d6. (arXiv, 2017). Near-optimal sample complexity for convex tensor completion, arXiv.","DOI":"10.1093\/imaiai\/iay019"},{"key":"ref_28","unstructured":"Ghadermarzy, N., Plan, Y., and Y\u0131lmaz, \u00d6. (arXiv, 2018). Learning tensors from partial binary measurements, arXiv."},{"key":"ref_29","unstructured":"Liu, Y., Shang, F., Fan, W., Cheng, J., and Cheng, H. (2014, January 8\u201313). Generalized Higher-Order Orthogonal Iteration for Tensor Decomposition and Completion. Proceedings of the Advances in Neural Information Processing Systems, Palais des Congr\u00e8s de Montr\u00e9al, Montr\u00e9al, Canada."},{"key":"ref_30","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_31","doi-asserted-by":"crossref","first-page":"1511","DOI":"10.1109\/TSP.2016.2639466","article-title":"Exact Tensor Completion Using t-SVD","volume":"65","author":"Zhang","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Sun, W., Chen, Y., Huang, L., and So, H.C. (2018). Tensor Completion via Generalized Tensor Tubal Rank Minimization using General Unfolding. IEEE Signal Process. Lett.","DOI":"10.1109\/LSP.2018.2819892"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1152","DOI":"10.1109\/TIP.2017.2762595","article-title":"Tensor Factorization for Low-Rank Tensor Completion","volume":"27","author":"Zhou","year":"2018","journal-title":"IEEE Trans. Image Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"V83","DOI":"10.1190\/geo2014-0467.1","article-title":"5D seismic data completion and denoising using a novel class of tensor decompositions","volume":"80","author":"Ely","year":"2015","journal-title":"Geophysics"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2411","DOI":"10.1109\/TMC.2015.2505729","article-title":"Adaptive Sampling of RF Fingerprints for Fine-grained Indoor Localization","volume":"15","author":"Liu","year":"2016","journal-title":"IEEE Trans. Mob. Comput."},{"key":"ref_36","unstructured":"Jiang, J.Q., and Ng, M.K. (arXiv, 2017). Exact Tensor Completion from Sparsely Corrupted Observations via Convex Optimization, arXiv."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Liu, X.Y., Aeron, S., Aggarwal, V., and Wang, X. (arXiv, 2016). Low-tubal-rank tensor completion using alternating minimization, arXiv.","DOI":"10.1117\/12.2224039"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/2200000016","article-title":"Distributed optimization and statistical learning via the alternating direction method of multipliers","volume":"3","author":"Boyd","year":"2011","journal-title":"Found. Trends Mach. Learn."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Zhou, P., and Feng, J. (2017, January 21\u201326). Outlier-Robust Tensor PCA. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA.","DOI":"10.1109\/CVPR.2017.419"},{"key":"ref_40","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. Opt."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1002\/nla.1845","article-title":"The power and Arnoldi methods in an algebra of circulants","volume":"20","author":"Gleich","year":"2013","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1109\/79.109206","article-title":"The tensor product: a mathematical programming language for FFTs and other fast DSP operations","volume":"9","author":"Granata","year":"1992","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/TIP.2003.819861","article-title":"Image quality assessment: from error visibility to structural similarity","volume":"13","author":"Wang","year":"2004","journal-title":"IEEE Trans. Image Process."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2886","DOI":"10.1109\/TIT.2015.2401574","article-title":"Simultaneously structured models with application to sparse and low-rank matrices","volume":"61","author":"Oymak","year":"2015","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"A3291","DOI":"10.1137\/15M101628X","article-title":"Scalable Robust Matrix Recovery: Frank-Wolfe Meets Proximal Methods","volume":"38","author":"Mu","year":"2016","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_46","unstructured":"Richard, E., Obozinski, G.R., and Vert, J.P. (2014, January 8\u201313). Tight convex relaxations for sparse matrix factorization. Proceedings of the Advances in Neural Information Processing Systems, Montreal, QC, Canada."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"5423","DOI":"10.1109\/TSP.2016.2586759","article-title":"Smooth PARAFAC Decomposition for Tensor Completion","volume":"64","author":"Yokota","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1109\/TPAMI.2013.164","article-title":"Simultaneous Tensor Decomposition and Completion Using Factor Priors","volume":"36","author":"Chen","year":"2014","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_49","unstructured":"Xutao Li, Y.Y., and Xu, X. (2017, January 4\u20139). Low-Rank Tensor Completion with Total Variation for Visual Data Inpainting. Proceedings of the AAAI Conference on Artificial Intelligence, San Francisco, CA, USA."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Yokota, T., Erem, B., Guler, S., Warfield, S.K., and Hontani, H. (arXiv, 2018). Missing Slice Recovery for Tensors Using a Low-rank Model in Embedded Space, arXiv.","DOI":"10.1109\/CVPR.2018.00861"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.neunet.2017.10.007","article-title":"Matrix completion by deep matrix factorization","volume":"98","author":"Fan","year":"2018","journal-title":"Neural Netw."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"A474","DOI":"10.1137\/110841229","article-title":"An Order-p Tensor Factorization with Applications in Imaging","volume":"35","author":"Martin","year":"2013","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_53","unstructured":"Liu, X.Y., and Wang, X. (arXiv, 2017). Fourth-order tensors with multidimensional discrete transforms, arXiv."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/11\/7\/94\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:10:38Z","timestamp":1760195438000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/11\/7\/94"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,28]]},"references-count":53,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2018,7]]}},"alternative-id":["a11070094"],"URL":"https:\/\/doi.org\/10.3390\/a11070094","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2018,6,28]]}}}