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Appl. Algebra Geometry"],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>We develop algebraic methods for computations with tensor data. We give three applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and amplification of low rank tensor structure. We introduce colored Brauer diagrams, which are used for algebraic computations and in analyzing their computational complexity. We present numerical experiments whose results show that the performance of the alternating least squares algorithm for rank 1 approximations for tensors can be improved using tensor amplification.<\/jats:p>","DOI":"10.1137\/19m1272494","type":"journal-article","created":{"date-parts":[[2021,1,4]],"date-time":"2021-01-04T13:32:19Z","timestamp":1609767139000},"page":"1-27","source":"Crossref","is-referenced-by-count":1,"title":["Algebraic Methods for Tensor Data"],"prefix":"10.1137","volume":"5","author":[{"given":"Neriman","family":"Tokcan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jonathan","family":"Gryak","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kayvan","family":"Najarian","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Harm","family":"Derksen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,1,4]]},"reference":[{"key":"atypb1","unstructured":"B. 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Wallach,\n                      Representations and Invariants of Classical Groups\n                      , Cambridge University Press, Cambridge, 1998."},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"A. Grothendieck,\n                      Produits tensoriels topologiques et espaces nucl\u00e9ares\n                      , Mem. Amer. Math. Soc. (1955).","DOI":"10.1090\/memo\/0016"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(90)90014-6"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"J. H\\aastad,\n                      Tensor Rank Is NP-Complete\n                      , Automata, Languages, and Programming (Stresa, 1989), Lecture Notes in Comput. Sci. 372, Springer, Berlin, 1989, pp. 451-460.","DOI":"10.1007\/BFb0035776"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1145\/2512329"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1002\/sapm192761164"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1002\/sapm19287139"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/07070111X"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1515\/math-2019-0028"},{"key":"atypb17","unstructured":"J. M. Landsberg,\n                      Tensors: Geometry and Applications\n                      , Grad. Stud. Math. 128, American Mathematical Society, Providence, RI, 2012."},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"S. Lang,\n                      Algebra\n                      , 3rd ed., Grad. 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Shafarevich, eds., Springer-Verlag, Berlin, 1994.","DOI":"10.1007\/978-3-662-03073-8_2"},{"key":"atypb25","unstructured":"L. Qi and S. Hu,\n                      Spectral Norm and Nuclear Norm of a Third Order Tensor\n                      , preprint, arXiv:1909.01529, 2019."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.26599\/BDMA.2018.9020008"},{"key":"atypb27","unstructured":"R. Schatten,\n                      A Theory of Cross-Spaces\n                      , Princeton University Press, Princeton, NJ, 1950."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2016.2570862"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1137\/06066518X"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/BF02289464"},{"key":"atypb31","unstructured":"H. Weyl,\n                      The Classical Groups. 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Williams,\n                      Invariant Polynomials on Tensors under the Action of a Product of Orthogonal Groups\n                      , Ph.D. Thesis, University of Wisconsin, Milwaukee, 2013."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-015-9269-5"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899352045"}],"container-title":["SIAM Journal on Applied Algebra and Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1272494","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:39:17Z","timestamp":1787233157000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1272494"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/19M1272494"],"URL":"https:\/\/doi.org\/10.1137\/19m1272494","relation":{},"ISSN":["2470-6566"],"issn-type":[{"value":"2470-6566","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}