{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:10Z","timestamp":1787333710686,"version":"build-2736575974"},"reference-count":64,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"name":"Hong Kong Research Grant Council","award":["PolyU153014\/18p"],"award-info":[{"award-number":["PolyU153014\/18p"]}]},{"DOI":"10.13039\/501100001459","name":"Ministry of Education, Singapore","doi-asserted-by":"crossref","award":["R-146-000-257-112"],"award-info":[{"award-number":["R-146-000-257-112"]}],"id":[{"id":"10.13039\/501100001459","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771328"],"award-info":[{"award-number":["11771328"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004517","name":"Tianjin University","doi-asserted-by":"publisher","award":["2017XZC-0084"],"award-info":[{"award-number":["2017XZC-0084"]}],"id":[{"id":"10.13039\/501100004517","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004517","name":"Tianjin University","doi-asserted-by":"publisher","award":["2017XRG-0015"],"award-info":[{"award-number":["2017XRG-0015"]}],"id":[{"id":"10.13039\/501100004517","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>In this paper, we study the polynomial optimization problem of a multiform over the intersection of the multisphere and the nonnegative orthants. This class of problems is NP-hard in general and includes the problem of finding the best nonnegative rank-one approximation of a given tensor. A Positivstellensatz is given for this class of polynomial optimization problems, based on which a globally convergent hierarchy of doubly nonnegative (DNN) relaxations is proposed. A (zeroth order) DNN relaxation method is applied to solve these problems, resulting in linear matrix optimization problems under both the positive semidefinite and nonnegative conic constraints. A worst case approximation bound is given for this relaxation method. The recent solver SDPNAL+ is adopted to solve this class of matrix optimization problems. Typically the DNN relaxations are tight, and hence the best nonnegative rank-one approximation of a tensor can be obtained frequently. Extensive numerical experiments show that this approach is quite promising.<\/jats:p>","DOI":"10.1137\/18m1224064","type":"journal-article","created":{"date-parts":[[2019,12,6]],"date-time":"2019-12-06T19:18:08Z","timestamp":1575659888000},"page":"1527-1554","source":"Crossref","is-referenced-by-count":4,"title":["Best Nonnegative Rank-One Approximations of Tensors"],"prefix":"10.1137","volume":"40","author":[{"given":"Shenglong","family":"Hu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Defeng","family":"Sun","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kim-Chuan","family":"Toh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,12,3]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2013.10.046"},{"key":"atypb2","unstructured":"N. Asgarian and R. Greiner,\n                      Using Rank-1 Biclusters to Classify Microarray Data\n                      , Department of Computing Science and the Alberta Ingenuity Center for Machine Learning, University of Alberta, Edmonton, Canada, 2006."},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"A. Ben-Tal and A. Nemirovskii,\n                      Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications\n                      , SIAM, Philadelphia, PA, 2001.","DOI":"10.1137\/1.9780898718829"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.67.031902"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"M. Biggs, A. Ghodsi, and S. Vavasis,\n                      Nonnegative matrix factorization via rank-one downdating\n                      , in Proceedings of the International Conference on Machine Learning, 2008.","DOI":"10.1145\/1390156.1390165"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"J. Bochnak, M. Coste, and M.F. Roy,\n                      Real Algebraic Geometry\n                      , Springer, Berlin, 1998.","DOI":"10.1007\/978-3-662-03718-8"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-128X(199709\/10)11:5<393::AID-CEM483>3.0.CO;2-L"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"A. Cichocki, R. Zdunek, A. Phan, and S. Amari,\n                      Nonnegative Matrix and Tensor Factorizations: Application to Exploratory Multi-Way Data Analysis and Blind Separation\n                      , Wiley, Hoboken, NJ, 2009.","DOI":"10.1002\/9780470747278"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90224-C"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"R. Curto and L. Fialkow,\n                      Solution of the Truncated Complex Moment Problem for Flat Data\n                      , Mem. Amer. Math. Soc. 119, American Mathematical Society, Providence, RI, 1996.","DOI":"10.1090\/memo\/0568"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479896305696"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479898346995"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2017.2697680"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-013-9594-z"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"D. FitzGerald, M. Cranitch, and E. Coyle,\n                      Non-negative tensor factorisation for sound source separation\n                      , in Proceedings of the Irish Signals and Systems Conference, 2005.","DOI":"10.1049\/cp:20050279"},{"key":"atypb16","first-page":"631","volume":"23","author":"Friedlander M.","year":"2008","journal-title":"Comput. Optim. Appl."},{"key":"atypb18","unstructured":"M. Garey and D. Johnson,\n                      Computers and Intractability, A Guide to the Theory of NP-Completeness\n                      , Freeman, New York, NY, 1980."},{"key":"atypb19","unstructured":"N. Gillis,\n                      Approximation et sous-approximation de matrices par factorisation positive: algorithmes, complexit\u00e9 et applications\n                      , Master's Thesis, Universit\u00e9 Catholique de Louvain, Louvain-la-Neuve, Belgium, 2006."},{"key":"atypb20","unstructured":"G. Hardy, J. Littlewood, and G. P\u00f3lya,\n                      Inequalities\n                      , 2nd ed., Cambridge University Press, Cambridge, UK, 1952."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(90)90014-6"},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"T. Hazan, S. Polak, and A. Shashua,\n                      Sparse image coding using a $3$D non-negative tensor factorization\n                      , in Proceedings of the 10th IEEE International Conference on Computer Vision, Vol. 1, IEEE Computer Society Press, 2005, pp. 50-57.","DOI":"10.1109\/ICCV.2005.228"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-018-0981-3"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2016.2576427"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479801387413"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/07070111X"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/100801482"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623400366802"},{"key":"atypb29","unstructured":"C. Lawson and R. Hanson,\n                      Solving Least Squares Problems\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1974."},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1038\/44565"},{"key":"atypb32","doi-asserted-by":"crossref","unstructured":"L.H. Lim,\n                      Tensors and hypermatrices\n                      , in Handbook of Linear Algebra, L. Hogben, ed., CRC Press, Boca Raton, FL 2013, Chapter 15.","DOI":"10.1201\/b16113-19"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1002\/cem.1244"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1137\/080729104"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1965-053-6"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1007\/BF02592948"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1007\/s11464-012-0187-4"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0680-x"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-005-0672-6"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/j.jco.2006.07.002"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1137\/130935112"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1137\/17M115308X"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1137\/15M1018514"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/S0169-7439(97)00031-2"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1002\/env.3170050203"},{"key":"atypb46","first-page":"83","author":"Parrilo P.","year":"2003","journal-title":"RI"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-014-0822-9"},{"key":"atypb48","first-page":"141","volume":"73","author":"P\u00f3lya G.","year":"1928","journal-title":"Naturforsch. Ges. Z\u00fcrich."},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(00)00155-9"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.1993.42.42045"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2013.03.015"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2016.2532906"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/253\/03936"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1016\/j.sigpro.2011.03.006"},{"key":"atypb55","doi-asserted-by":"crossref","unstructured":"A. Shashua and T. Hazan,\n                      Non-negative tensor factorization with applications to statistics and computer vision\n                      , in Proceedings of the 22nd International Conference on Machine Learning, 2005, pp. 792-799.","DOI":"10.1145\/1102351.1102451"},{"key":"atypb56","unstructured":"A. Shashua and A. Levin,\n                      Linear image coding for regression and classification using the tensor-rank principle\n                      , in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2001."},{"key":"atypb57","first-page":"595","author":"Shashua A.","year":"2006","journal-title":"Part"},{"key":"atypb58","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2017.2690524"},{"key":"atypb59","doi-asserted-by":"crossref","unstructured":"J. Sturm,\n                      SeDuMi 1.02: A Matlab toolbox for optimization over symmetric cones\n                      , Optim. Methods Softw., 11 & 12 (1999), pp. 625-653.","DOI":"10.1080\/10556789908805766"},{"key":"atypb60","doi-asserted-by":"publisher","DOI":"10.1137\/140964357"},{"key":"atypb62","doi-asserted-by":"publisher","DOI":"10.1080\/10556789908805762"},{"key":"atypb63","doi-asserted-by":"publisher","DOI":"10.1137\/070709967"},{"key":"atypb64","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8655(01)00070-8"},{"key":"atypb65","doi-asserted-by":"publisher","DOI":"10.1007\/s12532-015-0082-6"},{"key":"atypb66","doi-asserted-by":"publisher","DOI":"10.1137\/080718206"},{"key":"atypb67","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2015.2478396"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1224064","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:18:36Z","timestamp":1787332716000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1224064"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":64,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1224064"],"URL":"https:\/\/doi.org\/10.1137\/18m1224064","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}