{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T05:05:44Z","timestamp":1763960744002,"version":"3.45.0"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,8,15]],"date-time":"2025-08-15T00:00:00Z","timestamp":1755216000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,15]],"date-time":"2025-08-15T00:00:00Z","timestamp":1755216000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"European Union - PNRR","award":["N00000013 - CUP J33C22001170001"],"award-info":[{"award-number":["N00000013 - CUP J33C22001170001"]}]},{"DOI":"10.13039\/100031478","name":"NextGenerationEU","doi-asserted-by":"publisher","award":["20227PCCKZ \u2013 CUP J53D2300362000"],"award-info":[{"award-number":["20227PCCKZ \u2013 CUP J53D2300362000"]}],"id":[{"id":"10.13039\/100031478","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s11075-025-02204-8","type":"journal-article","created":{"date-parts":[[2025,8,15]],"date-time":"2025-08-15T04:31:29Z","timestamp":1755232289000},"page":"2069-2093","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["TT-LSQR for tensor least squares problems and application to data mining"],"prefix":"10.1007","volume":"100","author":[{"given":"Lorenzo","family":"Piccinini","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Valeria","family":"Simoncini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,8,15]]},"reference":[{"key":"2204_CR1","unstructured":"Algarte, R.D.: Tensor-based foundations of ordinary least squares and neural network regression models (2025). arXiv:2411.12873"},{"key":"2204_CR2","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1553\/etna_vol55s92","volume":"55","author":"AH Bentbib","year":"2022","unstructured":"Bentbib, A.H., Khouia, A., Sadok, H.: The LSQR method for solving tensor least-squares problems. Electron. Trans. Numer. Anal. 55, 92\u2013111 (2022)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"2204_CR3","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1137\/S106482750240649X","volume":"25","author":"M Benzi","year":"2003","unstructured":"Benzi, M., Tuma, M.: A robust preconditioner with low memory requirements for large sparse least squares problems. SIAM J. Sci. Comput. 25, 499\u2013512 (2003)","journal-title":"SIAM J. Sci. Comput."},{"key":"2204_CR4","volume-title":"Understanding Searching Engines, Software, Enviroments, and Tools","author":"MW Berry","year":"1999","unstructured":"Berry, M.W., Browne, M.: Understanding Searching Engines, Software, Enviroments, and Tools. SIAM, Philadelphia, USA (1999)"},{"key":"2204_CR5","doi-asserted-by":"publisher","first-page":"A1091","DOI":"10.1137\/24M1688540","volume":"47","author":"I Bioli","year":"2025","unstructured":"Bioli, I., Kressner, D., Robol, L.: Preconditioned low-rank riemannian optimization for symmetric positive definite linear matrix equations. SIAM J. Sci. Comput. 47, A1091\u2013A1116 (2025)","journal-title":"SIAM J. Sci. Comput."},{"key":"2204_CR6","doi-asserted-by":"crossref","unstructured":"Bouss\u00e9, M., Vervliet, N., Debals, O., De\u00a0Lathauwer, L.: Face recognition as a Kronecker product equation. In: IEEE 7th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), pp. 1\u20135 (2017)","DOI":"10.1109\/CAMSAP.2017.8313140"},{"key":"2204_CR7","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/s10092-020-0358-8","volume":"57","author":"D Brandoni","year":"2020","unstructured":"Brandoni, D., Simoncini, V.: Tensor-train decomposition for image recognition. Calcolo 57, 9 (2020)","journal-title":"Calcolo"},{"key":"2204_CR8","doi-asserted-by":"crossref","unstructured":"Bucci, A., Palitta, D., Robol, L.: Randomized sketched TT-GMRES for linear systems with tensor structure. Tech. Rep., To appear in SIAM J. Scient. Comput. (2024)","DOI":"10.1137\/24M1694999"},{"key":"2204_CR9","doi-asserted-by":"publisher","first-page":"2852","DOI":"10.1109\/TCYB.2023.3265279","volume":"54","author":"S-Y Chang","year":"2024","unstructured":"Chang, S.-Y., Wu, H.-C.: Tensor-based least-squares solutions for multirelational signals and applications. IEEE Trans. Cybern. 54, 2852\u20132865 (2024)","journal-title":"IEEE Trans. Cybern."},{"key":"2204_CR10","doi-asserted-by":"publisher","first-page":"1199","DOI":"10.1007\/s11075-019-00797-5","volume":"84","author":"M Chen","year":"2020","unstructured":"Chen, M., Kressner, D.: Recursive blocked algorithms for linear systems with Kronecker product structure. Numer. Algorithms 84, 1199\u20131216 (2020)","journal-title":"Numer. Algorithms"},{"key":"2204_CR11","doi-asserted-by":"crossref","unstructured":"Dantas, C.F., Cohen, J.E., Gribonval, R.: Learning tensor-structured dictionaries with application to hyperspectral image denoising. In: 27th European Signal Processing Conference, EUSIPCO 2019, A Coru\u00f1a, Spain, September 2\u20136, 2019. IEEE, pp. 1\u20135 (2019)","DOI":"10.23919\/EUSIPCO.2019.8902593"},{"key":"2204_CR12","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1109\/LSP.2017.2681159","volume":"24","author":"CF Dantas","year":"2017","unstructured":"Dantas, C.F., Da Costa, M.N., da\u00a0Rocha\u00a0Lopes, R.: Learning dictionaries as a sum of Kronecker products. IEEE Signal Process. Lett. 24, 559\u2013563 (2017)","journal-title":"IEEE Signal Process. Lett."},{"key":"2204_CR13","doi-asserted-by":"publisher","first-page":"1253","DOI":"10.1137\/S0895479896305696","volume":"21","author":"L De Lathauwer","year":"2000","unstructured":"De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253\u20131278 (2000)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"2204_CR14","doi-asserted-by":"publisher","first-page":"A2248","DOI":"10.1137\/140953289","volume":"36","author":"SV Dolgov","year":"2014","unstructured":"Dolgov, S.V., Savostyanov, D.V.: Alternating minimal energy methods for linear systems in higher dimensions. SIAM J. Sci. Comput. 36, A2248\u2013A2271 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"2204_CR15","doi-asserted-by":"crossref","unstructured":"Eld\u00e9n, L.: Matrix methods in data mining and pattern recognition, 2nd ed. Fundamentals of Algorithms, SIAM, Philadelphia, USA (2019)","DOI":"10.1137\/1.9781611975864"},{"key":"2204_CR16","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1515\/1569395054012767","volume":"13","author":"W Hackbusch","year":"2005","unstructured":"Hackbusch, W., Khoromskij, B.N., Tyrtyshnikov, E.E.: Hierarchical Kronecker tensor-product approximations. J. Numer. Math. 13, 119\u2013156 (2005)","journal-title":"J. Numer. Math."},{"key":"2204_CR17","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1137\/110842570","volume":"6","author":"N Hao","year":"2013","unstructured":"Hao, N., Kilmer, M.E., Braman, K., Hoover, R.C.: Facial recognition using tensor-tensor decompositions. SIAM J. Imaging Sci. 6, 437\u2013463 (2013)","journal-title":"SIAM J. Imaging Sci."},{"key":"2204_CR18","doi-asserted-by":"crossref","unstructured":"Howell, G.W., Baboulin, M.: LU preconditioning for overdetermined sparse least squares problems. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K., Kitowski, J., Wiatr, K. (eds.) Parallel Processing and Applied Mathematics, pp.\u00a0128\u2013137. Cham, Springer International Publishing (2016)","DOI":"10.1007\/978-3-319-32149-3_13"},{"key":"2204_CR19","doi-asserted-by":"publisher","first-page":"e2086","DOI":"10.1002\/nla.2086","volume":"24","author":"B Jiang","year":"2017","unstructured":"Jiang, B., Yang, F., Zhang, S.: Tensor and its Tucker core: the invariance relationships. Numer. Linear Algebra Appl. 24, e2086 (2017)","journal-title":"Numer. Linear Algebra Appl."},{"key":"2204_CR20","volume-title":"Applied multivariate statistical analysis","author":"RA Johnson","year":"2007","unstructured":"Johnson, R.A., Wichern, D.W.: Applied multivariate statistical analysis. Pearson Prentice Hall, NJ (2007)"},{"key":"2204_CR21","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1016\/j.laa.2015.07.021","volume":"485","author":"E Kernfeld","year":"2015","unstructured":"Kernfeld, E., Kilmer, M., Aeron, S.: Tensor-tensor products with invertible linear transforms. Linear Algebra Appl. 485, 545\u2013570 (2015)","journal-title":"Linear Algebra Appl."},{"key":"2204_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.chemolab.2011.09.001","volume":"110","author":"BN Khoromskij","year":"2012","unstructured":"Khoromskij, B.N.: Tensors-structured numerical methods in scientific computing: survey on recent advances. Chemom. Intell. Lab. Syst. 110, 1\u201319 (2012)","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"2204_CR23","doi-asserted-by":"publisher","DOI":"10.1515\/9783110365917","volume-title":"Tensor Numerical Methods in Scientific Computing","author":"BN Khoromskij","year":"2018","unstructured":"Khoromskij, B.N.: Tensor Numerical Methods in Scientific Computing. De Gruyter, Berlin, Boston (2018)"},{"key":"2204_CR24","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1137\/07070111X","volume":"51","author":"TG Kolda","year":"2009","unstructured":"Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51, 455\u2013500 (2009)","journal-title":"SIAM Rev."},{"key":"2204_CR25","doi-asserted-by":"publisher","first-page":"666","DOI":"10.1002\/nla.1919","volume":"21","author":"D Kressner","year":"2014","unstructured":"Kressner, D., Ple\u0161inger, M., Tobler, C.: A preconditioned low-rank CG method for parameter-dependent Lyapunov equations. Num. Lin. Alg. Appl. 21, 666\u2013684 (2014)","journal-title":"Num. Lin. Alg. Appl."},{"key":"2204_CR26","doi-asserted-by":"publisher","first-page":"1688","DOI":"10.1137\/090756843","volume":"31","author":"D Kressner","year":"2010","unstructured":"Kressner, D., Tobler, C.: Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl. 31, 1688\u20131714 (2010)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"2204_CR27","doi-asserted-by":"publisher","first-page":"1288","DOI":"10.1137\/100799010","volume":"32","author":"D Kressner","year":"2011","unstructured":"Kressner, D., Tobler, C.: Low-rank tensor Krylov subspace methods for parametrized linear systems. SIAM. J. Matrix Anal. & Appl. 32, 1288\u20131316 (2011)","journal-title":"SIAM. J. Matrix Anal. & Appl."},{"key":"2204_CR28","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1017\/S0962492920000021","volume":"29","author":"P-G Martinsson","year":"2020","unstructured":"Martinsson, P.-G., Tropp, J.A.: Randomized numerical linear algebra: Foundations and algorithms. Acta Numer. 29, 403\u2013572 (2020)","journal-title":"Acta Numer."},{"key":"2204_CR29","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1137\/23M1551973","volume":"45","author":"M Meier","year":"2024","unstructured":"Meier, M., Nakatsukasa, Y., Townsend, A., Webb, M.: Are sketch-and-precondition least squares solvers numerically stable? SIAM J. Matrix Anal. Appl. 45, 905\u2013929 (2024)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"2204_CR30","doi-asserted-by":"publisher","first-page":"2295","DOI":"10.1137\/090752286","volume":"33","author":"I Oseledets","year":"2011","unstructured":"Oseledets, I.: Tensor-train decomposition. SIAM J. Scientific Comput. 33, 2295\u20132317 (2011)","journal-title":"SIAM J. Scientific Comput."},{"key":"2204_CR31","unstructured":"Oseledets, I.: TT-Toolbox (2025). https:\/\/github.com\/oseledets\/TT-Toolbox"},{"key":"2204_CR32","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1016\/j.laa.2009.07.024","volume":"432","author":"I Oseledets","year":"2010","unstructured":"Oseledets, I., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432, 70\u201388 (2010)","journal-title":"Linear Algebra Appl."},{"key":"2204_CR33","doi-asserted-by":"publisher","first-page":"A2718","DOI":"10.1137\/110833142","volume":"34","author":"IV Oseledets","year":"2012","unstructured":"Oseledets, I.V., Dolgov, S.V.: Solution of linear systems and matrix inversion in the TT-format. SIAM J. Sci. Comput. 34, A2718\u2013A2739 (2012)","journal-title":"SIAM J. Sci. Comput."},{"key":"2204_CR34","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1145\/355984.355989","volume":"8","author":"CC Paige","year":"1982","unstructured":"Paige, C.C., Saunders, M.A.: LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. (TOMS) 8, 43\u201371 (1982)","journal-title":"ACM Trans. Math. Softw. (TOMS)"},{"key":"2204_CR35","doi-asserted-by":"publisher","first-page":"751","DOI":"10.1007\/s10543-015-0575-8","volume":"56","author":"D Palitta","year":"2016","unstructured":"Palitta, D., Simoncini, V.: Matrix-equation-based strategies for convection-diffusion equations. BIT Numer. Math. 56, 751\u2013776 (2016)","journal-title":"BIT Numer. Math."},{"key":"2204_CR36","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1007\/s10589-024-00629-w","volume":"91","author":"L Piccinini","year":"2025","unstructured":"Piccinini, L., Simoncini, V.: Truncated LSQR for matrix least squares problems. Comput. Optimization Appl. (COAP) 91, 905\u2013932 (2025)","journal-title":"Comput. Optimization Appl. (COAP)"},{"key":"2204_CR37","doi-asserted-by":"publisher","first-page":"13212","DOI":"10.1073\/pnas.0804869105","volume":"105","author":"V Rokhlin","year":"2008","unstructured":"Rokhlin, V., Tygert, M.: A fast randomized algorithm for overdetermined linear least-squares regression. Proc. Natl. Acad. Sci. U.S.A. 105, 13212\u201313217 (2008)","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2204_CR38","doi-asserted-by":"crossref","unstructured":"Saputra, N.A., Riza, L.S., Setiawan, A., Hamidah, I.: A systematic review for classification and selection of deep learning methods. Decision Anal. J. 12 (2024)","DOI":"10.1016\/j.dajour.2024.100489"},{"key":"2204_CR39","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1007\/s40574-020-00247-4","volume":"13","author":"V Simoncini","year":"2020","unstructured":"Simoncini, V.: Numerical solution of a class of third order tensor linear equations. Bollettino UMI 13, 429\u2013439 (2020)","journal-title":"Bollettino UMI"},{"key":"2204_CR40","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1137\/22M147880X","volume":"44","author":"V Simoncini","year":"2023","unstructured":"Simoncini, V., Hao, Y.: Analysis of the truncated conjugate gradient method for linear matrix equations. SIAM. J. Matrix Anal. & Appl. 44, 359\u2013381 (2023)","journal-title":"SIAM. J. Matrix Anal. & Appl."},{"key":"2204_CR41","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1561\/0400000060","volume":"10","author":"DP Woodruff","year":"2014","unstructured":"Woodruff, D.P.: Sketching as a tool for numerical linear algebra. Found. Trends Theor. Comput. Sci. 10, 1\u2013157 (2014)","journal-title":"Found. Trends Theor. Comput. Sci."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-025-02204-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-025-02204-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-025-02204-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T04:59:59Z","timestamp":1763960399000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-025-02204-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,15]]},"references-count":41,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["2204"],"URL":"https:\/\/doi.org\/10.1007\/s11075-025-02204-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2025,8,15]]},"assertion":[{"value":"2 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 August 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}