{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T15:32:53Z","timestamp":1787499173617,"version":"build-2736575974"},"reference-count":54,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["IIS-1837985"],"award-info":[{"award-number":["IIS-1837985"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2001460"],"award-info":[{"award-number":["DMS-2001460"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1638352"],"award-info":[{"award-number":["DMS-1638352"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1900460"],"award-info":[{"award-number":["CCF-1900460"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001782","name":"University of Melbourne","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001782","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Algebra Geometry"],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>We study the log-likelihood function and maximum likelihood estimate (MLE) for the matrix normal model for both real and complex models. We describe the exact number of samples needed to achieve (almost surely) three conditions, namely a bounded log-likelihood function, existence of MLEs, and uniqueness of MLEs. As a consequence, we observe that almost sure boundedness of the log-likelihood function guarantees almost sure existence of an MLE, thereby proving a conjecture of Drton, Kuriki, and Hoff [ Existence and Uniqueness of the Kronecker Covariance MLE, preprint, arXiv:2003.06024, 2020]. The main tools we use are from the theory of quiver representations, in particular, results of Kac, King, and Schofield on canonical decomposition and stability.<\/jats:p>","DOI":"10.1137\/20m1369348","type":"journal-article","created":{"date-parts":[[2021,6,21]],"date-time":"2021-06-21T12:39:21Z","timestamp":1624279161000},"page":"338-365","source":"Crossref","is-referenced-by-count":20,"title":["Maximum Likelihood Estimation for Matrix Normal Models via Quiver Representations"],"prefix":"10.1137","volume":"5","author":[{"given":"Harm","family":"Derksen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Visu","family":"Makam","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,6,21]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1214\/09-AOAS314"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/20M1328932"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"A. Borel,\n                      Linear Algebraic Groups\n                      , 2nd ed., Grad. Texts Math. 126, Springer-Verlag, New York, 1991.","DOI":"10.1007\/978-1-4612-0941-6"},{"key":"atypb4","first-page":"1","author":"Borel A.","year":"2006","journal-title":"RI"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.2307\/1970210"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2005.04.015"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.2307\/1970884"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s00209-006-0008-0"},{"key":"atypb9","first-page":"845","author":"B\u00fcrgisser P.","year":"2019","journal-title":"MD"},{"key":"atypb10","unstructured":"C. Chindris and D. Kline,\n                      Simultaneous Robust Subspace Recovery and Semi-stability of Quiver Representations\n                      , preprint,https:\/\/arxiv.org\/abs\/2003.02962, 2020."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2002.1011197"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2017.01.018"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2017.06.035"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2017.12.007"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/jdm043"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-00-00331-3"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1023\/A:1020007100426"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"H. Derksen and J. Weyman,\n                      An Introduction to Quiver Representations\n                      , Grad. Stud. Math. 184, American Mathematical Society, Providence, RI, 2017.","DOI":"10.1090\/gsm\/184"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-10-00662-4"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01236060"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"M. Drton, S. Kuriki, and P. Hoff,\n                      Existence and Uniqueness of the Kronecker Covariance MLE\n                      , preprint,https:\/\/arxiv.org\/abs\/2003.06024, 2020.","DOI":"10.1214\/21-AOS2052"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1080\/00949659908811970"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-095X(199611)7:6<551::AID-ENV233>3.0.CO;2-9"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176350372"},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"A. Garg, L. Gurvits, R. Oliveira, and A. Widgerson,\n                      A deterministic polynomial time algorithm for non-commutative rational identity testing\n                      , in Proceedings of the 2016 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS), New Brunswick, NJ, 2016, pp. 109-117.","DOI":"10.1109\/FOCS.2016.95"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-018-0434-2"},{"key":"atypb27","first-page":"145","author":"Ginzburg V.","year":"2012","journal-title":"S\u00e9min. Congr. 24-I, Soc. Math. France, Paris"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcss.2004.06.003"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/BF01208503"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/BF01444162"},{"key":"atypb31","unstructured":"V. Hoskins,\n                      Geometric Invariant Theory and Symplectic Quotients\n                      , manuscript,http:\/\/userpage.fu-berlin.de\/hoskins\/GITnotes.pdf, 2012."},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1109\/TBME.2002.1001967"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-016-0143-x"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-018-0165-7"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7152(87)90078-2"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1007\/BF01403155"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(82)90105-3"},{"key":"atypb38","first-page":"76","author":"Keller B.","year":"2010","journal-title":"UK"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/45.4.515"},{"key":"atypb40","doi-asserted-by":"crossref","unstructured":"G. D. Mostow,\n                      Self-adjoint groups\n                      , Ann. of Math. (2), 62 (1955), pp. 44-55.","DOI":"10.2307\/2007099"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1090\/jams\/864"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2005.04.020"},{"key":"atypb43","first-page":"347","volume":"6","author":"Mardia K. V.","year":"1993","journal-title":"Multivariate Environ. Statist."},{"key":"atypb44","first-page":"589","author":"Reineke M.","year":"2008","journal-title":"Z\u00fcrich"},{"key":"atypb45","first-page":"87","volume":"22","author":"Ressayre N.","year":"2011","journal-title":"S.)"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1007\/BF02411903"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2015.05.019"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-65.1.46"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.3103\/S1066530708040066"},{"key":"atypb50","first-page":"125","volume":"12","author":"Schofield A.","year":"2001","journal-title":"S.)"},{"key":"atypb51","unstructured":"T. A. Springer,\n                      Linear Algebraic Groups\n                      , reprint of the 1998 2nd ed., Modern Birkh\u00e4user Classics, Birkh\u00e4user Boston, Inc., Boston, MA, 2009."},{"key":"atypb52","unstructured":"B. Torr\u00e9sani and E. Villaron,\n                      Harmonic hidden Markov models for the study of EEG signals\n                      , in Proceedings of the 2010 18th European Signal Processing Conference (EUSIPCO-2010), Aalborg, Denmark, 2010, pp. 711-715."},{"key":"atypb53","first-page":"2550","volume":"23","author":"Zhang Y.","year":"2010","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2012.2218241"}],"container-title":["SIAM Journal on Applied Algebra and Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1369348","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:32:08Z","timestamp":1787229128000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1369348"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":54,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1369348"],"URL":"https:\/\/doi.org\/10.1137\/20m1369348","relation":{},"ISSN":["2470-6566"],"issn-type":[{"value":"2470-6566","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}