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Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We present a new framework for computing $Z$-eigenvectors of general tensors based on numerically integrating a dynamical system that can only converge to a $Z$-eigenvector. Our motivation comes from our recent research on spacey random walks, where the long-term dynamics of a stochastic process are governed by a dynamical system that must converge to a $Z$-eigenvector of a transition probability tensor. Here, we apply the ideas more broadly to general tensors and find that our method can compute $Z$-eigenvectors that algebraic methods like the higher-order power method cannot compute.<\/jats:p>","DOI":"10.1137\/18m1229584","type":"journal-article","created":{"date-parts":[[2019,11,12]],"date-time":"2019-11-12T11:00:35Z","timestamp":1573556435000},"page":"1311-1324","source":"Crossref","is-referenced-by-count":18,"title":["Computing Tensor $Z$-Eigenvectors with Dynamical Systems"],"prefix":"10.1137","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6110-1583","authenticated-orcid":true,"given":"Austin R.","family":"Benson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8107-6474","authenticated-orcid":true,"given":"David F.","family":"Gleich","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,11,12]]},"reference":[{"key":"atypb1","first-page":"2773","volume":"15","author":"Anandkumar A.","year":"2014","journal-title":"J. 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Leskovec (2015),\n                      Tensor spectral clustering for partitioning higher-order network structures\n                      , in Proceedings of the 2015 SIAM International Conference on Data Mining, Vancouver, British Columbia, Canada, pp. 118-126,https:\/\/doi.org\/10.1137\/1.9781611974010.14.","DOI":"10.1137\/1.9781611974010.14"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/16M1074023"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1002\/nla.809"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2011.05.040"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/0605020"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/060661569"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/140962292"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/140985160"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2005.01.009"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1145\/2512329"},{"key":"atypb16","doi-asserted-by":"crossref","first-page":"012304","DOI":"10.1103\/PhysRevA.93.012304","volume":"93","author":"Hu S.","year":"2016","journal-title":"Physical Rev. A, 93, https:\/\/doi.org\/10.1103\/physreva."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/17M1133312"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/280\/04625"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479801387413"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/07070111X"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/100801482"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/140951758"},{"key":"atypb23","unstructured":"L. De Lathauwer (1997),\n                      Signal Processing Based on Multilinear Algebra\n                      , Ph.D. thesis, Katholieke Universiteit Leuven, Leuven, Belgium."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479898346995"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1080\/03081087.2013.777436"},{"key":"atypb26","unstructured":"L.H. Lim (2005),\n                      Singular values and eigenvalues of tensors: A variational approach\n                      , in 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Puerto Vallarta, Mexico, pp. 129-132."},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"L.H. Lim (2013),\n                      Tensors and hypermatrices\n                      , in Handbook of Linear Algebra, 2nd ed., L. Hogben, ed., Chapman and Hall\/CRC Press, Boca Raton, FL, pp. 231-260.","DOI":"10.1201\/b16113-19"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/100796765"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"B. Meini and F. Poloni (2018),\n                      Perron-based algorithms for the multilinear PageRank\n                      , Numer. 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Kofidis (2000),\n                      The higher-order power method revisited: Convergence proofs and effective initialization\n                      , in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, IEEE, Istanbul, Turkey,https:\/\/doi.org\/10.1109\/icassp.2000.861047.","DOI":"10.1109\/ICASSP.2000.861047"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.68.042307"},{"key":"atypb37","unstructured":"T. Wu, A. Benson, and D. F. Gleich (2016),\n                      General tensor spectral co-clustering for higher-order data\n                      , in Advances in Neural Information Processing Systems, pp. 2559-2567, Barcelona, Spain,http:\/\/papers.nips.cc\/paper\/6376-general-tensor-spectral-co-clustering-for-higher-order-data."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1229584","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:17:35Z","timestamp":1787332655000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1229584"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":37,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1229584"],"URL":"https:\/\/doi.org\/10.1137\/18m1229584","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}