{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:42:43Z","timestamp":1787330563286,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1419018"],"award-info":[{"award-number":["DMS-1419018"]}],"id":[{"id":"10.13039\/100000001","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>Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry, and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor.<\/jats:p>","DOI":"10.1137\/18m1212331","type":"journal-article","created":{"date-parts":[[2019,4,5]],"date-time":"2019-04-05T12:17:33Z","timestamp":1554466653000},"page":"394-416","source":"Crossref","is-referenced-by-count":18,"title":["Learning Paths from Signature Tensors"],"prefix":"10.1137","volume":"40","author":[{"given":"Max","family":"Pfeffer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2407-1095","authenticated-orcid":true,"given":"Anna","family":"Seigal","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6642-1479","authenticated-orcid":true,"given":"Bernd","family":"Sturmfels","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,4,3]]},"reference":[{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01086737"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2017.10.009"},{"key":"atypb4","first-page":"1455","volume":"15","author":"Boumal N.","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"P. B\u00fcrgisser and F. Cucker,\n                      Condition: The Geometry of Numerical Algorithms\n                      , Grundlehren Math. Wiss. 349, Springer-Verlag, Berlin, 2013.","DOI":"10.1007\/978-3-642-38896-5"},{"key":"atypb6","unstructured":"P. B\u00fcrgisser, A. Garg, R. Oliveira, M. Walter, and A. Wigderson,\n                      Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory\n                      , arXiv:1711.08039, 2017."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.2307\/1969671"},{"key":"atypb8","unstructured":"I. Chevyrev and A. Kormilitzin,\n                      A Primer on the Signature Method in Machine Learning\n                      , arXiv:1603.03788, 2016."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01400115"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"J. Diehl and J. Reizenstein,\n                      Invariants of Multidimensional Time Series Based on their Iterated-Integral Signature\n                      , arXiv:1801.06104, 2018.","DOI":"10.1007\/s10440-018-00227-z"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-014-9240-x"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"P. Friz and M. Hairer,\n                      A Course on Rough Paths. With an Introduction to Regularity Structures\n                      , Universitext, Springer-Verlag, Berlin, 2014.","DOI":"10.1007\/978-3-319-08332-2"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"F. Galuppi,\n                      The Rough Veronese Variety\n                      , arXiv:1809.02522, 2018.","DOI":"10.1016\/j.laa.2019.08.029"},{"key":"atypb14","unstructured":"S. Garibaldi and R. Guralnick,\n                      Generically Free Representations II: Irreducible Representations\n                      , arXiv:1711.06400, 2017."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/07070111X"},{"key":"atypb17","unstructured":"A. Kormilitzin, K. Saunders, P. Harrison, J. Geddes and T. Lyons,\n                      Application of the Signature Method to Pattern Recognition in the CEQUEL Clinical Trial\n                      , arXiv:1606.02074, 2016."},{"key":"atypb18","unstructured":"A. Kormilitzin, K. Saunders, P. Harrison, J. Geddes, and T. Lyons,\n                      Detecting Early Signs of Depressive and Manic Episodes in Patients with Bipolar Disorder using the Signature-Based Model\n                      , arXiv:1708.01206, 2017."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(77)90069-6"},{"key":"atypb20","unstructured":"J. M. Landsberg,\n                      Tensors: Geometry and Applications\n                      , Grad. Stud. Math. 128, AMS, Providence, RI, 2012."},{"key":"atypb21","unstructured":"D. Levin, T. Lyons, and H. Ni,\n                      Learning from the Past, Predicting the Statistics for the Future, Learning an Evolving System\n                      , arXiv:1309.0260, 2018."},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"T. Lyons and Z. Qian,\n                      System Control and Rough Paths\n                      , Oxford University Press, Oxford, UK, 2002.","DOI":"10.1093\/acprof:oso\/9780198506485.001.0001"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2016.12.024"},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"T. Lyons and W. Xu,\n                      Inverting the signature of a path\n                      , J. Eur. Math. Soc (JEMS), 20 (2018), pp. 1655-1687.","DOI":"10.4171\/jems\/796"},{"key":"atypb25","unstructured":"J. Nocedal and S. J. Wright,\n                      Numerical Optimization\n                      , Springer-Verlag, Berlin, 2006."},{"key":"atypb26","first-page":"3","volume":"48","author":"Popov A. M.","year":"1988","journal-title":"Trans. Moscow Math. Soc."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/0805026"},{"key":"atypb28","doi-asserted-by":"crossref","unstructured":"C. Reutenauer,\n                      Free Lie Algebras\n                      , London Math. Soc. Monogr. 7, Oxford University Press, New York, 1993.","DOI":"10.1093\/oso\/9780198536796.001.0001"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/BF02289464"},{"key":"atypb30","first-page":"406","author":"Strassen V.","year":"1987","journal-title":"J. Reine Angew. Math., ("},{"key":"atypb31","unstructured":"N. Vervliet, O. Debals, L. Sorber, M. Van Barel, and L. De Lathauwer,\n                      Tensorlab\n                      3.0,https:\/\/www.tensorlab.net(2016)."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1212331","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:12:16Z","timestamp":1787328736000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1212331"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1212331"],"URL":"https:\/\/doi.org\/10.1137\/18m1212331","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}