{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T04:27:45Z","timestamp":1782448065424,"version":"3.54.5"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,2,19]],"date-time":"2021-02-19T00:00:00Z","timestamp":1613692800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,2,19]],"date-time":"2021-02-19T00:00:00Z","timestamp":1613692800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2021,3]]},"DOI":"10.1007\/s00211-021-01185-9","type":"journal-article","created":{"date-parts":[[2021,2,20]],"date-time":"2021-02-20T00:36:03Z","timestamp":1613781363000},"page":"727-757","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["An approximation method of CP rank for third-order tensor completion"],"prefix":"10.1007","volume":"147","author":[{"given":"Chao","family":"Zeng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tai-Xiang","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael K.","family":"Ng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,2,19]]},"reference":[{"issue":"1","key":"1185_CR1","first-page":"2116","volume":"18","author":"M Ashraphijuo","year":"2017","unstructured":"Ashraphijuo, M., Wang, X.: Fundamental conditions for low-CP-rank tensor completion. J. Mach. Learn. Res. 18(1), 2116\u20132145 (2017)","journal-title":"J. Mach. Learn. Res."},{"issue":"2","key":"1185_CR2","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1287\/moor.1100.0449","volume":"35","author":"H Attouch","year":"2010","unstructured":"Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-\u0141ojasiewicz inequality. Math. Oper. Res. 35(2), 438\u2013457 (2010)","journal-title":"Math. Oper. Res."},{"key":"1185_CR3","unstructured":"Bader, B.W., Kolda, T.G. et\u00a0al.: MATLAB Tensor Toolbox Version 3.0-dev. https:\/\/www.tensortoolbox.org (2017)"},{"key":"1185_CR4","unstructured":"Barak, B., Moitra, A.: Noisy tensor completion via the sum-of-squares hierarchy. In: Conference on Learning Theory, pp. 417\u2013445 (2016)"},{"issue":"1\u20132","key":"1185_CR5","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1007\/s10107-013-0701-9","volume":"146","author":"J Bolte","year":"2014","unstructured":"Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1\u20132), 459\u2013494 (2014)","journal-title":"Math. Program."},{"issue":"1","key":"1185_CR6","first-page":"1","volume":"3","author":"S Boyd","year":"2011","unstructured":"Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends\u00ae Mach. Learn. 3(1), 1\u2013122 (2011)","journal-title":"Found. Trends\u00ae Mach. Learn."},{"issue":"3","key":"1185_CR7","doi-asserted-by":"publisher","first-page":"2435","DOI":"10.1137\/17M114618X","volume":"28","author":"P Breiding","year":"2018","unstructured":"Breiding, P., Vannieuwenhoven, N.: A Riemannian trust region method for the canonical tensor rank approximation problem. SIAM J. Optim. 28(3), 2435\u20132465 (2018)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1185_CR8","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1137\/17M1142880","volume":"39","author":"P Breiding","year":"2018","unstructured":"Breiding, P., Vannieuwenhoven, N.: The condition number of join decompositions. SIAM J. Matrix Anal. Appl. 39(1), 287\u2013309 (2018)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"4","key":"1185_CR9","doi-asserted-by":"publisher","first-page":"1956","DOI":"10.1137\/080738970","volume":"20","author":"J-F Cai","year":"2010","unstructured":"Cai, J.-F., Cand\u00e8s, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956\u20131982 (2010)","journal-title":"SIAM J. Optim."},{"issue":"6","key":"1185_CR10","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1007\/s10208-009-9045-5","volume":"9","author":"EJ Cand\u00e8s","year":"2009","unstructured":"Cand\u00e8s, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9(6), 717 (2009)","journal-title":"Found. Comput. Math."},{"issue":"5","key":"1185_CR11","doi-asserted-by":"publisher","first-page":"2053","DOI":"10.1109\/TIT.2010.2044061","volume":"56","author":"EJ Cand\u00e8s","year":"2010","unstructured":"Cand\u00e8s, E.J., Tao, T.: The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inf. Theory 56(5), 2053\u20132080 (2010)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1185_CR12","unstructured":"Cocosco, C.A., Kollokian, V., Kwan, R.K.-S., Pike, G.B., Evans, A.C.: Brainweb: Online interface to a 3D MRI simulated brain database. In NeuroImage, Citeseer (1997)"},{"issue":"4","key":"1185_CR13","doi-asserted-by":"publisher","first-page":"1253","DOI":"10.1137\/S0895479896305696","volume":"21","author":"L De Lathauwer","year":"2000","unstructured":"De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253\u20131278 (2000)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"1185_CR14","doi-asserted-by":"publisher","first-page":"1084","DOI":"10.1137\/06066518X","volume":"30","author":"V De Silva","year":"2008","unstructured":"De Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084\u20131127 (2008)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"2","key":"1185_CR15","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1137\/S0895479895290954","volume":"20","author":"A Edelman","year":"1998","unstructured":"Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303\u2013353 (1998)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"311","key":"1185_CR16","doi-asserted-by":"publisher","first-page":"1255","DOI":"10.1090\/mcom\/3239","volume":"87","author":"S Friedland","year":"2018","unstructured":"Friedland, S., Lim, L.-H.: Nuclear norm of higher-order tensors. Math. Comput. 87(311), 1255\u20131281 (2018)","journal-title":"Math. Comput."},{"issue":"2","key":"1185_CR17","doi-asserted-by":"publisher","first-page":"025010","DOI":"10.1088\/0266-5611\/27\/2\/025010","volume":"27","author":"S Gandy","year":"2011","unstructured":"Gandy, S., Recht, B., Yamada, I.: Tensor completion and low-n-rank tensor recovery via convex optimization. Inverse Prob. 27(2), 025010 (2011)","journal-title":"Inverse Prob."},{"issue":"1","key":"1185_CR18","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1137\/130905010","volume":"35","author":"D Goldfarb","year":"2014","unstructured":"Goldfarb, D., Qin, Z.: Robust low-rank tensor recovery: models and algorithms. SIAM J. Matrix Anal. Appl. 35(1), 225\u2013253 (2014)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"4","key":"1185_CR19","doi-asserted-by":"publisher","first-page":"644","DOI":"10.1016\/0196-6774(90)90014-6","volume":"11","author":"J H\u00e5stad","year":"1990","unstructured":"H\u00e5stad, J.: Tensor rank is NP-complete. J. Algorithms 11(4), 644\u2013654 (1990)","journal-title":"J. Algorithms"},{"issue":"6","key":"1185_CR20","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1145\/2512329","volume":"60","author":"CJ Hillar","year":"2013","unstructured":"Hillar, C.J., Lim, L.-H.: Most tensor problems are NP-hard. J. ACM (JACM) 60(6), 45 (2013)","journal-title":"J. ACM (JACM)"},{"issue":"2","key":"1185_CR21","doi-asserted-by":"publisher","first-page":"A683","DOI":"10.1137\/100818893","volume":"34","author":"S Holtz","year":"2012","unstructured":"Holtz, S., Rohwedder, T., Schneider, R.: The alternating linear scheme for tensor optimization in the tensor train format. SIAM J. Sci. Comput. 34(2), A683\u2013A713 (2012)","journal-title":"SIAM J. Sci. Comput."},{"key":"1185_CR22","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139020411","volume-title":"Matrix Analysis","author":"RA Horn","year":"2012","unstructured":"Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)"},{"key":"1185_CR23","unstructured":"Jain, P., Oh, S.: Provable tensor factorization with missing data. In: Advances in Neural Information Processing Systems, pp. 1431\u20131439 (2014)"},{"issue":"2","key":"1185_CR24","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1007\/s10107-014-0774-0","volume":"150","author":"B Jiang","year":"2015","unstructured":"Jiang, B., Ma, S., Zhang, S.: Tensor principal component analysis via convex optimization. Math. Program. 150(2), 423\u2013457 (2015)","journal-title":"Math. Program."},{"issue":"6","key":"1185_CR25","doi-asserted-by":"publisher","first-page":"1390","DOI":"10.1109\/JSTSP.2018.2873144","volume":"12","author":"B Jiang","year":"2018","unstructured":"Jiang, B., Ma, S., Zhang, S.: Low-M-rank tensor completion and robust tensor PCA. IEEE J. Sel. Top. Signal Process. 12(6), 1390\u20131404 (2018)","journal-title":"IEEE J. Sel. Top. Signal Process."},{"issue":"3","key":"1185_CR26","doi-asserted-by":"publisher","first-page":"e2086","DOI":"10.1002\/nla.2086","volume":"24","author":"B Jiang","year":"2017","unstructured":"Jiang, B., Yang, F., Zhang, S.: Tensor and its Tucker core: the invariance relationships. Numer. Linear Algebra Appl. 24(3), e2086 (2017)","journal-title":"Numer. Linear Algebra Appl."},{"key":"1185_CR27","doi-asserted-by":"crossref","unstructured":"Jiang, Q., Ng, M.: Robust low-tubal-rank tensor completion via convex optimization. In: Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, pp. 2649\u20132655 (2019)","DOI":"10.24963\/ijcai.2019\/368"},{"issue":"1","key":"1185_CR28","doi-asserted-by":"publisher","first-page":"148","DOI":"10.1137\/110837711","volume":"34","author":"ME Kilmer","year":"2013","unstructured":"Kilmer, M.E., Braman, K., Hao, N., Hoover, R.C.: Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging. SIAM J. Matrix Anal. Appl. 34(1), 148\u2013172 (2013)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"1185_CR29","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1137\/07070111X","volume":"51","author":"TG Kolda","year":"2009","unstructured":"Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455\u2013500 (2009)","journal-title":"SIAM Rev."},{"issue":"2","key":"1185_CR30","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1007\/s10543-013-0455-z","volume":"54","author":"D Kressner","year":"2014","unstructured":"Kressner, D., Steinlechner, M., Vandereycken, B.: Low-rank tensor completion by Riemannian optimization. BIT Numer. Math. 54(2), 447\u2013468 (2014)","journal-title":"BIT Numer. Math."},{"issue":"2","key":"1185_CR31","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/0024-3795(77)90069-6","volume":"18","author":"JB Kruskal","year":"1977","unstructured":"Kruskal, J.B.: Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra Appl. 18(2), 95\u2013138 (1977)","journal-title":"Linear Algebra Appl."},{"key":"1185_CR32","volume-title":"Tensors: Geometry and Applications","author":"JM Landsberg","year":"2012","unstructured":"Landsberg, J.M.: Tensors: Geometry and Applications, vol. 128. American Mathematical Society, Providence (2012)"},{"issue":"1","key":"1185_CR33","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1109\/TPAMI.2012.39","volume":"35","author":"J Liu","year":"2013","unstructured":"Liu, J., Musialski, P., Wonka, P., Ye, J.: Tensor completion for estimating missing values in visual data. IEEE Trans. Pattern Anal. Mach. Intell. 35(1), 208\u2013220 (2013)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"1185_CR34","unstructured":"Mu, C., Huang, B., Wright, J., Goldfarb, D.: Square deal: lower bounds and improved relaxations for tensor recovery. In: International conference on machine learning, pp. 73\u201381 (2014)"},{"issue":"3","key":"1185_CR35","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1137\/070697835","volume":"52","author":"B Recht","year":"2010","unstructured":"Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471\u2013501 (2010)","journal-title":"SIAM Rev."},{"issue":"2","key":"1185_CR36","doi-asserted-by":"publisher","first-page":"1134","DOI":"10.1137\/110857520","volume":"51","author":"T Rohwedder","year":"2013","unstructured":"Rohwedder, T., Uschmajew, A.: On local convergence of alternating schemes for optimization of convex problems in the tensor train format. SIAM J. Numer. Anal. 51(2), 1134\u20131162 (2013)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"1185_CR37","doi-asserted-by":"publisher","first-page":"1142","DOI":"10.1109\/SURV.2011.082911.00067","volume":"14","author":"P Seeling","year":"2011","unstructured":"Seeling, P., Reisslein, M.: Video transport evaluation with H. 264 video traces. IEEE Commun. Surv. Tutor. 14(4), 1142\u20131165 (2011)","journal-title":"IEEE Commun. Surv. Tutor."},{"issue":"5","key":"1185_CR38","doi-asserted-by":"publisher","first-page":"S461","DOI":"10.1137\/15M1010506","volume":"38","author":"M Steinlechner","year":"2016","unstructured":"Steinlechner, M.: Riemannian optimization for high-dimensional tensor completion. SIAM J. Sci. Comput. 38(5), S461\u2013S484 (2016)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"1185_CR39","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1137\/110843587","volume":"33","author":"A Uschmajew","year":"2012","unstructured":"Uschmajew, A.: Local convergence of the alternating least squares algorithm for canonical tensor approximation. SIAM J. Matrix Anal. Appl. 33(2), 639\u2013652 (2012)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1185_CR40","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/j.laa.2017.08.014","volume":"535","author":"N Vannieuwenhoven","year":"2017","unstructured":"Vannieuwenhoven, N.: Condition numbers for the tensor rank decomposition. Linear Algebra Appl. 535, 35\u201386 (2017)","journal-title":"Linear Algebra Appl."},{"issue":"1\u20132","key":"1185_CR41","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1007\/s10107-012-0584-1","volume":"142","author":"Z Wen","year":"2013","unstructured":"Wen, Z., Yin, W.: A feasible method for optimization with orthogonality constraints. Math. Program. 142(1\u20132), 397\u2013434 (2013)","journal-title":"Math. Program."},{"issue":"4","key":"1185_CR42","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1007\/s12532-012-0044-1","volume":"4","author":"Z Wen","year":"2012","unstructured":"Wen, Z., Yin, W., Zhang, Y.: Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm. Math. Program. Comput. 4(4), 333\u2013361 (2012)","journal-title":"Math. Program. Comput."},{"issue":"1","key":"1185_CR43","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s10107-015-0892-3","volume":"151","author":"SJ Wright","year":"2015","unstructured":"Wright, S.J.: Coordinate descent algorithms. Math. Program. 151(1), 3\u201334 (2015)","journal-title":"Math. Program."},{"issue":"2","key":"1185_CR44","doi-asserted-by":"publisher","first-page":"601","DOI":"10.3934\/ipi.2015.9.601","volume":"9","author":"Y Xu","year":"2015","unstructured":"Xu, Y., Hao, R., Yin, W., Su, Z.: Parallel matrix factorization for low-rank tensor completion. Inverse Problems Imag. 9(2), 601\u2013624 (2015)","journal-title":"Inverse Problems Imag."},{"issue":"3","key":"1185_CR45","doi-asserted-by":"publisher","first-page":"1758","DOI":"10.1137\/120887795","volume":"6","author":"Y Xu","year":"2013","unstructured":"Xu, Y., Yin, W.: A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM J. Imag. Sci. 6(3), 1758\u20131789 (2013)","journal-title":"SIAM J. Imag. Sci."},{"issue":"281","key":"1185_CR46","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1090\/S0025-5718-2012-02598-1","volume":"82","author":"J Yang","year":"2013","unstructured":"Yang, J., Yuan, X.: Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization. Math. Comput. 82(281), 301\u2013329 (2013)","journal-title":"Math. Comput."},{"issue":"1","key":"1185_CR47","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1137\/140983689","volume":"26","author":"Y Yang","year":"2016","unstructured":"Yang, Y., Feng, Y., Huang, X., Suykens, J.A.: Rank-1 tensor properties with applications to a class of tensor optimization problems. SIAM J. Optim. 26(1), 171\u2013196 (2016)","journal-title":"SIAM J. Optim."},{"issue":"20","key":"1185_CR48","doi-asserted-by":"publisher","first-page":"5423","DOI":"10.1109\/TSP.2016.2586759","volume":"64","author":"T Yokota","year":"2016","unstructured":"Yokota, T., Zhao, Q., Cichocki, A.: Smooth PARAFAC decomposition for tensor completion. IEEE Trans. Signal Process. 64(20), 5423\u20135436 (2016)","journal-title":"IEEE Trans. Signal Process."},{"issue":"4","key":"1185_CR49","doi-asserted-by":"publisher","first-page":"1031","DOI":"10.1007\/s10208-015-9269-5","volume":"16","author":"M Yuan","year":"2016","unstructured":"Yuan, M., Zhang, C.-H.: On tensor completion via nuclear norm minimization. Found. Comput. Math. 16(4), 1031\u20131068 (2016)","journal-title":"Found. Comput. Math."},{"issue":"6","key":"1185_CR50","doi-asserted-by":"publisher","first-page":"1511","DOI":"10.1109\/TSP.2016.2639466","volume":"65","author":"Z Zhang","year":"2017","unstructured":"Zhang, Z., Aeron, S.: Exact tensor completion using t-SVD. IEEE Trans. Signal Process. 65(6), 1511\u20131526 (2017)","journal-title":"IEEE Trans. Signal Process."},{"issue":"9","key":"1185_CR51","doi-asserted-by":"publisher","first-page":"1751","DOI":"10.1109\/TPAMI.2015.2392756","volume":"37","author":"Q Zhao","year":"2015","unstructured":"Zhao, Q., Zhang, L., Cichocki, A.: Bayesian CP factorization of incomplete tensors with automatic rank determination. IEEE Trans. Pattern Anal. Mach. Intell. 37(9), 1751\u20131763 (2015)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-021-01185-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-021-01185-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-021-01185-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,29]],"date-time":"2023-01-29T12:04:05Z","timestamp":1674993845000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-021-01185-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,19]]},"references-count":51,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,3]]}},"alternative-id":["1185"],"URL":"https:\/\/doi.org\/10.1007\/s00211-021-01185-9","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,19]]},"assertion":[{"value":"4 October 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 August 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 January 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 February 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}