{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T16:08:20Z","timestamp":1786118900260,"version":"build-2736575974"},"reference-count":63,"publisher":"American Mathematical Society (AMS)","issue":"362","license":[{"start":{"date-parts":[[2027,3,11]],"date-time":"2027-03-11T00:00:00Z","timestamp":1804723200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["2022NBN7TL"],"award-info":[{"award-number":["2022NBN7TL"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["663281"],"award-info":[{"award-number":["663281"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["STG\/19\/002"],"award-info":[{"award-number":["STG\/19\/002"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["C16\/21\/002"],"award-info":[{"award-number":["C16\/21\/002"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["663281"],"award-info":[{"award-number":["663281"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["2022NBN7TL"],"award-info":[{"award-number":["2022NBN7TL"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["663281"],"award-info":[{"award-number":["663281"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["STG\/19\/002"],"award-info":[{"award-number":["STG\/19\/002"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["C16\/21\/002"],"award-info":[{"award-number":["C16\/21\/002"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["663281"],"award-info":[{"award-number":["663281"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004497","name":"Onderzoeksraad, KU Leuven","doi-asserted-by":"publisher","award":["2022NBN7TL"],"award-info":[{"award-number":["2022NBN7TL"]}],"id":[{"id":"10.13039\/501100004497","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004497","name":"Onderzoeksraad, KU Leuven","doi-asserted-by":"publisher","award":["663281"],"award-info":[{"award-number":["663281"]}],"id":[{"id":"10.13039\/501100004497","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004497","name":"Onderzoeksraad, KU Leuven","doi-asserted-by":"publisher","award":["STG\/19\/002"],"award-info":[{"award-number":["STG\/19\/002"]}],"id":[{"id":"10.13039\/501100004497","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004497","name":"Onderzoeksraad, KU Leuven","doi-asserted-by":"publisher","award":["C16\/21\/002"],"award-info":[{"award-number":["C16\/21\/002"]}],"id":[{"id":"10.13039\/501100004497","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004497","name":"Onderzoeksraad, KU Leuven","doi-asserted-by":"publisher","award":["663281"],"award-info":[{"award-number":["663281"]}],"id":[{"id":"10.13039\/501100004497","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["2022NBN7TL"],"award-info":[{"award-number":["2022NBN7TL"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["663281"],"award-info":[{"award-number":["663281"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["STG\/19\/002"],"award-info":[{"award-number":["STG\/19\/002"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["C16\/21\/002"],"award-info":[{"award-number":["C16\/21\/002"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["663281"],"award-info":[{"award-number":["663281"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>A Hadamard\u2013Hitchcock decomposition of a multidimensional array is a decomposition that expresses the latter as a Hadamard product of several tensor rank decompositions. Such decompositions can encode probability distributions that arise from statistical graphical models associated to complete bipartite graphs with one layer of observed random variables and one layer of hidden ones, usually called restricted Boltzmann machines. We establish generic identifiability of Hadamard\u2013Hitchcock decompositions by exploiting the reshaped Kruskal criterion for tensor rank decompositions. An algorithm leveraging existing decomposition algorithms for tensor rank decomposition is introduced for computing a Hadamard\u2013Hitchcock decomposition. Numerical experiments illustrate its computational performance and numerical accuracy in a noiseless setting.<\/p>","DOI":"10.1090\/mcom\/4174","type":"journal-article","created":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T18:40:01Z","timestamp":1773254401000},"page":"2849-2890","source":"Crossref","is-referenced-by-count":0,"title":["Hadamard\u2013Hitchcock decompositions: Identifiability and computation"],"prefix":"10.1090","volume":"95","author":[{"given":"Alessandro","family":"Oneto","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nick","family":"Vannieuwenhoven","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2026,3,11]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"Paper No. 107568, 22","DOI":"10.1016\/j.jpaa.2023.107568","article-title":"Hadamard products and binomial ideals","volume":"228","author":"Atar, B\u00fc\u015fra","year":"2024","journal-title":"J. Pure Appl. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4049","issn-type":"print"},{"issue":"6","key":"2","doi-asserted-by":"publisher","first-page":"3099","DOI":"10.1214\/09-AOS689","article-title":"Identifiability of parameters in latent structure models with many observed variables","volume":"37","author":"Allman, Elizabeth S.","year":"2009","journal-title":"Ann. Statist.","ISSN":"https:\/\/id.crossref.org\/issn\/0090-5364","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"767","DOI":"10.1090\/S0002-9947-08-04725-9","article-title":"Induction for secant varieties of Segre varieties","volume":"361","author":"Abo, Hirotachi","year":"2009","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"739","DOI":"10.1137\/18M1200531","article-title":"Pencil-based algorithms for tensor rank decomposition are not stable","volume":"40","author":"Beltr\u00e1n, Carlos","year":"2019","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/s10208-022-09551-1","article-title":"The average condition number of most tensor rank decomposition problems is infinite","volume":"23","author":"Beltr\u00e1n, Carlos","year":"2023","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"key":"6","series-title":"Unitext","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-24624-2","volume-title":"An introduction to algebraic statistics with tensors","volume":"118","author":"Bocci, Cristiano","year":"2019","ISBN":"https:\/\/id.crossref.org\/isbn\/9783030246235"},{"key":"7","doi-asserted-by":"publisher","first-page":"Paper No. 107590, 60","DOI":"10.1016\/j.aim.2021.107590","article-title":"Permutations, tensor products, and Cuntz algebra automorphisms","volume":"381","author":"Brenti, Francesco","year":"2021","journal-title":"Adv. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"issue":"11","key":"8","doi-asserted-by":"publisher","first-page":"Paper No. 107078, 12","DOI":"10.1016\/j.jpaa.2022.107078","article-title":"Hadamard products of hypersurfaces","volume":"226","author":"Bocci, Cristiano","year":"2022","journal-title":"J. Pure Appl. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4049","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"[BCC{\\etalchar{+}}18] A. Bernardi, E. Carlini, M. V. Catalisano, A. Gimigliano, and A. Oneto, The hitchhiker guide to: secant varieties and tensor decomposition, Mathematics 6 (2018), no. 12, 314.","DOI":"10.3390\/math6120314"},{"key":"10","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1016\/j.jalgebra.2015.10.008","article-title":"Hadamard products of linear spaces","volume":"448","author":"Bocci, Cristiano","year":"2016","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"11","series-title":"Information Science and Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-45528-0","volume-title":"Pattern recognition and machine learning","author":"Bishop, Christopher M.","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387310732"},{"key":"12","doi-asserted-by":"crossref","unstructured":"[BPA{\\etalchar{+}}21] M. Besan\u00e7on, T. Papamarkou, D. Anthoff, A. Arslan, S. Byrne, D. Lin, and J. Pearson, Distributions.jl: definition and modeling of probability distributions in the JuliaStats ecosystem, J. Statist. Softw. 98 (2021), no. 16, 1\u201330.","DOI":"10.18637\/jss.v098.i16"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1137\/17M1142880","article-title":"The condition number of join decompositions","volume":"39","author":"Breiding, Paul","year":"2018","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"2435","DOI":"10.1137\/17M114618X","article-title":"A Riemannian trust region method for the canonical tensor rank approximation problem","volume":"28","author":"Breiding, Paul","year":"2018","journal-title":"SIAM J. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/1052-6234","issn-type":"print"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"123","DOI":"10.4171\/pm\/2029","article-title":"On the Hadamard product of degenerate subvarieties","volume":"76","author":"Calussi, Gabriele","year":"2019","journal-title":"Port. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0032-5155","issn-type":"print"},{"issue":"4","key":"16","doi-asserted-by":"publisher","first-page":"2306","DOI":"10.1007\/s10618-024-01033-y","article-title":"The Hadamard decomposition problem","volume":"38","author":"Ciaperoni, Martino","year":"2024","journal-title":"Data Min. Knowl. Discov.","ISSN":"https:\/\/id.crossref.org\/issn\/1384-5810","issn-type":"print"},{"key":"17","isbn-type":"print","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1090\/conm\/516\/10172","article-title":"Geometry of the restricted Boltzmann machine","author":"Cueto, Mar\u00eda Ang\u00e9lica","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821848913"},{"issue":"3","key":"18","doi-asserted-by":"publisher","first-page":"1018","DOI":"10.1137\/110829180","article-title":"On generic identifiability of 3-tensors of small rank","volume":"33","author":"Chiantini, Luca","year":"2012","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"19","doi-asserted-by":"crossref","unstructured":"[COV14] L. Chiantini, G. Ottaviani, and N. Vannieuwenhoven, On generic and low-rank specific identifiability of complex tensors, SIAM Journal on Matrix Analysis and Applications, 35 (2014), no. 4, 1265\u20131287.","DOI":"10.1137\/140961389"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"656","DOI":"10.1137\/16M1090132","article-title":"Effective criteria for specific identifiability of tensors and forms","volume":"38","author":"Chiantini, Luca","year":"2017","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"issue":"12","key":"21","doi-asserted-by":"publisher","first-page":"1296","DOI":"10.1016\/j.jsc.2010.06.011","article-title":"An implicitization challenge for binary factor analysis","volume":"45","author":"Cueto, Mar\u00eda Ang\u00e9lica","year":"2010","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"22","doi-asserted-by":"crossref","unstructured":"[CZPA09] A. Cichocki, R. Zdunek, A.-H. Phan, and S.-I. Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation, John Wiley & Sons, Ltd., 2009.","DOI":"10.1002\/9780470747278"},{"key":"23","doi-asserted-by":"crossref","unstructured":"[DJL25] Z. Ding, T. Ji, and M. Li, Improved Hadamard decomposition and its application in data compression in new-type power systems, Mathematics, 13 (2025), no. 4, 671.","DOI":"10.3390\/math13040671"},{"issue":"3","key":"24","doi-asserted-by":"publisher","first-page":"1084","DOI":"10.1137\/06066518X","article-title":"Tensor rank and the ill-posedness of the best low-rank approximation problem","volume":"30","author":"de Silva, Vin","year":"2008","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"issue":"4","key":"25","doi-asserted-by":"publisher","first-page":"1427","DOI":"10.1007\/s00211-024-01427-6","article-title":"Which constraints of a numerical problem cause ill-conditioning?","volume":"156","author":"Dewaele, Nick","year":"2024","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"3","key":"26","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1007\/s00365-009-9076-9","article-title":"Black box low tensor-rank approximation using fiber-crosses","volume":"30","author":"Espig, Mike","year":"2009","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"key":"27","unstructured":"[FI12] A. Fischer and C. Igel, An introduction to restricted Boltzmann machines, Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications: 17th Iberoamerican Congress, CIARP 2012, Buenos Aires, Argentina, September 3\u20136, 2012, Proceedings 17, Springer, 2012, pp. 14\u201336."},{"key":"28","isbn-type":"print","first-page":"133","article-title":"Minkowski sums and Hadamard products of algebraic varieties","author":"Friedenberg, Netanel","year":"2017","ISBN":"https:\/\/id.crossref.org\/isbn\/9781493974856"},{"issue":"2","key":"29","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1007\/s11565-013-0175-y","article-title":"An asymptotic bound for secant varieties of Segre varieties","volume":"59","author":"Gesmundo, Fulvio","year":"2013","journal-title":"Ann. Univ. Ferrara Sez. VII Sci. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0430-3202","issn-type":"print"},{"key":"30","series-title":"Data Science","isbn-type":"print","volume-title":"Nonnegative matrix factorization","volume":"2","author":"Gillis, Nicolas","year":"[2021] \\copyright2021","ISBN":"https:\/\/id.crossref.org\/isbn\/9781611976403"},{"key":"31","series-title":"Universitext","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-9425-9","volume-title":"Multilinear algebra","author":"Greub, Werner","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0387902848","edition":"2"},{"issue":"3","key":"32","doi-asserted-by":"publisher","first-page":"Art. 12, 25","DOI":"10.1145\/1356052.1356053","article-title":"Anatomy of high-performance matrix multiplication","volume":"34","author":"Goto, Kazushige","year":"2008","journal-title":"ACM Trans. Math. Software","ISSN":"https:\/\/id.crossref.org\/issn\/0098-3500","issn-type":"print"},{"key":"33","series-title":"Johns Hopkins Studies in the Mathematical Sciences","isbn-type":"print","volume-title":"Matrix computations","author":"Golub, Gene H.","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9781421407944","edition":"4"},{"key":"34","unstructured":"[Har70] R. A. Harshman, Foundations of the PARAFAC procedure: models and conditions for an \u201cexplanatory\u201d multi-modal factor analysis, UCLA Working Papers in Phonetics, vol. 16, 1970, pp. 1\u201384."},{"key":"35","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2189-8","volume-title":"Algebraic geometry","volume":"133","author":"Harris, Joe","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0387977163"},{"key":"36","isbn-type":"print","volume-title":"Accuracy and stability of numerical algorithms","author":"Higham, Nicholas J.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713552"},{"key":"37","doi-asserted-by":"crossref","unstructured":"[Hit27] F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of products, J. Math. Phys. 6 (1927), no. 1-4, 164\u2013189.","DOI":"10.1002\/sapm192761164"},{"key":"38","doi-asserted-by":"crossref","unstructured":"[Ho95] T. K. Ho, Random decision forests, Proceedings of 3rd International Conference on Document Analysis and Recognition, ICDAR-95, vol. 1, IEEE Comput. Soc. Press, 1995, pp. 278\u2013282.","DOI":"10.1109\/ICDAR.1995.598994"},{"issue":"7","key":"39","doi-asserted-by":"publisher","first-page":"1527","DOI":"10.1162\/neco.2006.18.7.1527","article-title":"A fast learning algorithm for deep belief nets","volume":"18","author":"Hinton, Geoffrey E.","year":"2006","journal-title":"Neural Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0899-7667","issn-type":"print"},{"issue":"3","key":"40","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1137\/07070111X","article-title":"Tensor decompositions and applications","volume":"51","author":"Kolda, Tamara G.","year":"2009","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"2","key":"41","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/0024-3795(77)90069-6","article-title":"Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics","volume":"18","author":"Kruskal, Joseph B.","year":"1977","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"key":"42","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/128","volume-title":"Tensors: geometry and applications","volume":"128","author":"Landsberg, J. M.","year":"2012","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821869079"},{"key":"43","series-title":"Oxford Statistical Science Series","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198522195.001.0001","volume-title":"Graphical models","volume":"17","author":"Lauritzen, Steffen L.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0198522193"},{"key":"44","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1017\/S0962492921000076","article-title":"Tensors in computations","volume":"30","author":"Lim, Lek-Heng","year":"2021","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"4","key":"45","doi-asserted-by":"publisher","first-page":"1064","DOI":"10.1137\/0614071","article-title":"A decomposition for three-way arrays","volume":"14","author":"Leurgans, S. E.","year":"1993","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"46","unstructured":"[LWB{\\etalchar{+}}19] D. Lin, J. M. White, S. Byrne, D. Bates, A. Noack, J. Pearson, A. Arslan, K. Squire, D. Anthoff, T. Papamarkou, M. Besan\u00b8on, J. Drugowitsch, M. Schauer, and other contributors, JuliaStats\/Distributions.jl: a Julia package for probability distributions and associated functions, July 2019."},{"key":"47","first-page":"653","article-title":"Discrete restricted Boltzmann machines","volume":"16","author":"Mont\u00fafar, Guido","year":"2015","journal-title":"J. Mach. Learn. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"issue":"1","key":"48","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1137\/16M1077489","article-title":"Dimension of marginals of Kronecker product models","volume":"1","author":"Mont\u00fafar, Guido","year":"2017","journal-title":"SIAM J. Appl. Algebra Geom."},{"issue":"17","key":"49","doi-asserted-by":"publisher","first-page":"3293","DOI":"10.1215\/00127094-2023-0072","article-title":"Bronowski\u2019s conjecture and the identifiability of projective varieties","volume":"173","author":"Massarenti, Alex","year":"2024","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"50","doi-asserted-by":"crossref","unstructured":"[Mon16] G. Mont\u00fafar, Restricted Boltzmann machines: introduction and review, Information Geometry and its Applications IV, Springer, 2016, pp. 75\u2013115.","DOI":"10.1007\/978-3-319-97798-0_4"},{"key":"51","doi-asserted-by":"crossref","unstructured":"[PFS16] E. E. Papalexakis, C. Faloutsos, and N. D. Sidiropoulos, Tensors for data mining and data fusion: models, applications, and scalable algorithms, ACM Trans. Intell. Syst. Technol. 8 (2016), no. 16, 1\u201344.","DOI":"10.1145\/2915921"},{"issue":"7","key":"52","doi-asserted-by":"publisher","first-page":"1818","DOI":"10.1016\/j.laa.2009.11.033","article-title":"A concise proof of Kruskal\u2019s theorem on tensor decomposition","volume":"432","author":"Rhodes, John A.","year":"2010","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"issue":"1","key":"53","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18409\/jas.v9i1.90","article-title":"Mixtures and products in two graphical models","volume":"9","author":"Seigal, Anna","year":"2018","journal-title":"J. Algebr. Stat."},{"key":"54","unstructured":"[Smo86] P. Smolensky, Information processing in dynamical systems: foundations of harmony theory, Technical Report, Department of Computer Science, University of Colorado, Boulder, 1986."},{"key":"55","isbn-type":"print","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1090\/pspum\/062.2\/1492542","article-title":"Equations defining toric varieties","author":"Sturmfels, Bernd","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0821808958"},{"key":"56","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/194","volume-title":"Algebraic statistics","volume":"194","author":"Sullivant, Seth","year":"2018","ISBN":"https:\/\/id.crossref.org\/isbn\/9781470435172"},{"issue":"6","key":"57","doi-asserted-by":"publisher","first-page":"Paper No. e2446, 21","DOI":"10.1002\/nla.2446","article-title":"Tensor completion using geodesics on Segre manifolds","volume":"29","author":"Swijsen, Lars","year":"2022","journal-title":"Numer. Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1070-5325","issn-type":"print"},{"key":"58","unstructured":"[TBC23] A. Taveira Blomenhofer and A. Casarotti, Nondefectivity of invariant secant varieties,  arXiv:2312.12335v2, 2023."},{"issue":"4","key":"59","doi-asserted-by":"publisher","first-page":"Art. 38, 35","DOI":"10.1145\/3555369","article-title":"A normal form algorithm for tensor rank decomposition","volume":"48","author":"Telen, Simon","year":"2022","journal-title":"ACM Trans. Math. Software","ISSN":"https:\/\/id.crossref.org\/issn\/0098-3500","issn-type":"print"},{"key":"60","doi-asserted-by":"crossref","unstructured":"[VC71] V. N. Vapnik and A. Ya. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability & Its Applications, 16 (1971), no. 2, 264\u2013280.","DOI":"10.1137\/1116025"},{"issue":"2","key":"61","doi-asserted-by":"publisher","first-page":"A1027--A1052","DOI":"10.1137\/110836067","article-title":"A new truncation strategy for the higher-order singular value decomposition","volume":"34","author":"Vannieuwenhoven, Nick","year":"2012","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"62","doi-asserted-by":"crossref","first-page":"613","DOI":"10.2140\/pjm.1967.23.613","article-title":"Transformations on tensor spaces","volume":"23","author":"Westwick, Roy","year":"1967","journal-title":"Pacific J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0030-8730","issn-type":"print"},{"issue":"11","key":"63","doi-asserted-by":"publisher","first-page":"2817","DOI":"10.1016\/j.automatica.2008.04.009","article-title":"A dynamical systems approach to weighted graph matching","volume":"44","author":"Zavlanos, Michael M.","year":"2008","journal-title":"Automatica J. IFAC","ISSN":"https:\/\/id.crossref.org\/issn\/0005-1098","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-362\/S0025-5718-2026-04174-2\/mcom4174_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-362\/S0025-5718-2026-04174-2\/S0025-5718-2026-04174-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T17:15:52Z","timestamp":1786036552000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-362\/S0025-5718-2026-04174-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,11]]},"references-count":63,"journal-issue":{"issue":"362","published-print":{"date-parts":[[2026,11]]}},"alternative-id":["S0025-5718-2026-04174-2"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4174","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2026,3,11]]}}}