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Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>Hankel tensors are generalizations of Hankel matrices. This article studies both the computational and algebraic aspects of Hankel tensor ranks. We prove that for a low rank symmetric tensor, there exists a base change to make it a Hankel tensor. We also provide an algorithm that can compute the Vandermonde ranks and decompositions for all Hankel tensors. For a generic $n$-dimensional Hankel tensor of even order or order three, we prove that the candecomp-parafac rank, symmetric rank, Vandermonde rank, border rank, symmetric border rank, and Vandermonde border rank all coincide with each other. Some open questions are also posed.<\/jats:p>","DOI":"10.1137\/18m1168285","type":"journal-article","created":{"date-parts":[[2019,5,2]],"date-time":"2019-05-02T10:58:36Z","timestamp":1556794716000},"page":"486-516","source":"Crossref","is-referenced-by-count":11,"title":["Hankel Tensor Decompositions and Ranks"],"prefix":"10.1137","volume":"40","author":[{"given":"Jiawang","family":"Nie","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3193-7264","authenticated-orcid":true,"given":"Ke","family":"Ye","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,5,2]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2012.05.012"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2010.08.001"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2010.06.046"},{"key":"atypb4","unstructured":"P. B\u00fcrgisser, M. Clausen, and M. A. Shokrollahi,\n                      Algebraic Complexity Theory\n                      , Grundlehren Math. Wiss. 315, Springer, New York, 2013."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2015.11.051"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-015-0155-8"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/6762"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-010-9077-x"},{"key":"atypb9","unstructured":"P. Comon,\n                      Tensor decompositions\n                      , in Proceedings of Mathematics in Signal Processing V, (2002), pp. 1-24."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/060661569"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/140962292"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479896305696"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/S089547980139786X"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/06066518X"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-016-0622-0"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/130916084"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3239"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.201310004"},{"key":"atypb20","unstructured":"D. R. Grayson and M. E. Stillman,\n                      Macaulay 2, A Software System for Research in Algebraic Geometry\n                      ,http:\/\/www.vivc.edu\/MQcaulay2(2002)."},{"key":"atypb21","unstructured":"J. Harris,\n                      Algebraic Geometry: A First Course\n                      , Grad. Texts in Math. 133, Springer, New York, 2013."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(90)90014-6"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1145\/2512329"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1002\/sapm192761164"},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"A. Iarrobino and V. Kanev,\n                      Power Sums, Gorenstein Algebras, and Determinantal Varieties\n                      , Lecture Notes in Math., 1721, Springer, New York, 1999.","DOI":"10.1007\/BFb0093426"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/090764827"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/07070111X"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(77)90069-6"},{"key":"atypb29","unstructured":"J. M. Landsberg,\n                      Tensors: Geometry and Applications\n                      , Grad. Stud. Math. 128, AMS, Providence, RI, 2012."},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-003-0115-9"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2015.v011a011"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-009-9055-3"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/bdm049"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1137\/0614071"},{"key":"atypb35","first-page":"129","author":"Lim L.-H.","year":"2005","journal-title":"Multi-Sensor Adaptive Processing, IEEE"},{"key":"atypb36","doi-asserted-by":"crossref","unstructured":"L.H. Lim,\n                      Tensors and hypermatrices\n                      ,\n                      in\n                      Handbook of Linear Algebra, L. Hogben, ed., Chapman and Hall\/CRC Press, London, 2013.","DOI":"10.1201\/b16113-19"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/j.crme.2010.06.005"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2002.801933"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-015-9291-7"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1137\/16M1107528"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1137\/130935112"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-017-9973-y"},{"key":"atypb43","unstructured":"L. Oeding,\n                      Report on \u201cGeometry and Representation Theory of Tensors for Computer Science, Statistics and Other Areas,''\n                      preprint,arXiv:0810.3940, 2008."},{"key":"atypb44","unstructured":"L. Oeding,\n                      Border Ranks of Monomials\n                      , preprint,arXiv:1608.02530, 2016."},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2012.11.005"},{"key":"atypb46","unstructured":"G. Ottaviani,\n                      Symplectic bundles on the plane, secant varieties and L\u00fcroth quartics revisited\n                      , in Vector Bundles and Low Codimensional Subvarieties: State of the Art and Recent Developments, Quad. Mat. 21, G. Casnati, F. Catanese, and R. Notari, eds., Aracne, Trento, Italy, 2008, pp. 315-352."},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1002\/nla.453"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2005.05.007"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.02.071"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2015.v13.n1.a6"},{"key":"atypb51","unstructured":"Y. Shitov,\n                      How Hard Is the Tensor Rank?\n                      , preprint,arXiv:1611.01559, 2016."},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1137\/17M1131970"},{"key":"atypb53","unstructured":"J. J. Sylvester,\n                      An essay on canonical forms, supplement to a sketch of a memoir on elimination\n                      , in Collected Mathematical Papers Vol. 1, Chelsea, New York, 1973."},{"key":"atypb54","unstructured":"J. J. Sylvester,\n                      On a remarkable discovery in the theory of canonical forms and of hyperdeterminants\n                      , in Collected Mathematical Papers Vol. 1, Chelsea, New York, 1973."},{"key":"atypb55","first-page":"3634","author":"Trickett S.","year":"2013","journal-title":"Society of Exploration Geophysicists"},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-016-9332-x"},{"key":"atypb57","unstructured":"K. Ye and L.H. Lim,\n                      Tensor Network Ranks\n                      , preprint,arXiv:1801.02662, 2018."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1168285","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:13:16Z","timestamp":1787328796000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1168285"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":56,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1168285"],"URL":"https:\/\/doi.org\/10.1137\/18m1168285","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}