{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:25:31Z","timestamp":1787333131710,"version":"build-2736575974"},"reference-count":55,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>In this paper we proposed quasi-Newton and limited memory quasi-Newton methods for objective functions defined on Grassmannians or a product of Grassmannians. Specifically we defined BFGS and limited memory BFGS updates in local and global coordinates on Grassmannians or a product of these. We proved that, when local coordinates are used, our BFGS updates on Grassmannians share the same optimality property as the usual BFGS updates on Euclidean spaces. When applied to the best multilinear rank approximation problem for general and symmetric tensors, our approach yields fast, robust, and accurate algorithms that exploit the special Grassmannian structure of the respective problems and which work on tensors of large dimensions and arbitrarily high order. Extensive numerical experiments are included to substantiate our claims.<\/jats:p>","DOI":"10.1137\/090763172","type":"journal-article","created":{"date-parts":[[2010,11,30]],"date-time":"2010-11-30T18:33:55Z","timestamp":1291142035000},"page":"3352-3393","source":"Crossref","is-referenced-by-count":88,"title":["Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors"],"prefix":"10.1137","volume":"32","author":[{"given":"Berkant","family":"Savas","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lek-Heng","family":"Lim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,11,30]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-005-0179-9"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"P.A. Absil, R. Mahony, and R. Sepulchre,\n                      Optimization Algorithms on Matrix Manifolds\n                      , Princeton University Press, Princeton, NJ, 2008.","DOI":"10.1515\/9781400830244"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"E. Anderson, Z. Bai, C. H. Bischof, J. W. Demmel, J. J. Dongarra, J. J. Du Croz, A. Greenbaum, S. J. Hammarling, A. McKenney, S. Ostrouchov, and D. C. Sorensen,\n                      LAPACK Users' Guide\n                      , 3rd ed., SIAM, Philadelphia, 1999.","DOI":"10.1137\/1.9780898719604"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1145\/1186785.1186794"},{"key":"R5","unstructured":"W. M. Boothby,\n                      An Introduction to Differentiable Manifolds and Riemannian Geometry\n                      , 2nd ed., Academic Press, Orlando, FL, 1986."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01582063"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479896302898"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/060661569"},{"key":"R9","first-page":"125","volume":"31","author":"Comon P.","year":"2006","journal-title":"in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '06)"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/0165-1684(96)00079-5"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"L. Conlon,\n                      Differentiable Manifolds\n                      , 2nd ed., Birkh\u00e4user, Boston, MA, 2001.","DOI":"10.1007\/978-0-8176-4767-4"},{"key":"R12","unstructured":"L. De Lathauwer,\n                      Tucker compression, parallel factor analysis and block term decompositions: New results\n                      , European Meeting on Challenges in Modern Massive Data Sets (EMMDS '09), Technical University of Denmark, Copenhagen, Denmark, 2009."},{"key":"R13","unstructured":"L. De Lathauwer,\n                      Signal Processing Based on Multilinear Algebra\n                      , Ph.D. thesis, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, 1997."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1002\/1099-128X(200005\/06)14:3<123::AID-CEM589>3.0.CO;2-1"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479896305696"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479898346995"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2004.01.016"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30110-3_38"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/06066518X"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/1019005"},{"key":"R21","first-page":"167","volume":"4","author":"Dennis J. E.","year":"1981","journal-title":"Nonlinear Program."},{"key":"R22","doi-asserted-by":"crossref","unstructured":"J. E. Dennis and R. B. Schnabel,\n                      Numerical Methods for Unconstrained Optimization and Nonlinear Equations\n                      , SIAM, Philadelphia, 1996.","DOI":"10.1137\/1.9781611971200"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479895290954"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/070688316"},{"key":"R25","first-page":"1","volume":"4","author":"Flad H.-J.","year":"2008","journal-title":"Russian J. Numer. Anal. Math. Modelling"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1007\/BF00934767"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1976-0423804-2"},{"key":"R28","unstructured":"G. H. Golub and C. F. Van Loan,\n                      Matrix Computations\n                      , 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996."},{"key":"R29","doi-asserted-by":"crossref","unstructured":"W. Greub,\n                      Multilinear Algebra\n                      , 2nd ed., Springer-Verlag, New York, NY, 1978.","DOI":"10.1007\/978-1-4613-9425-9"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1002\/sapm192761164"},{"key":"R31","first-page":"337","volume":"42","author":"Ishteva M.","year":"2008","journal-title":"Int. J. Pure Appl. Math."},{"key":"R32","doi-asserted-by":"crossref","unstructured":"M. Ishteva, L. De Lathauwer, P.A. Absil, and S. Van Huffel,\n                      Best Low Multilinear Rank Approximation of Higher-order Tensors, Based on the Riemannian Trust-region Scheme\n                      , Technical report 09-142, ESAT-SISTA, Katholieke Universiteit Leuven, Leuven, Belgium, 2009.","DOI":"10.1007\/978-3-642-12598-0_13"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-008-9251-2"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.2478\/s11533-007-0018-0"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1137\/080730408"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479801387413"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1137\/07070111X"},{"key":"R38","unstructured":"S. Lang,\n                      Algebra\n                      , rev. 3rd ed., Springer-Verlag, New York, 2002."},{"key":"R39","doi-asserted-by":"crossref","unstructured":"R. Lehoucq, D. Sorensen, and C. Yang,\n                      ARPACK Users' Guide\n                      , SIAM, Philadelphia, 1998.","DOI":"10.1137\/1.9780898719628"},{"key":"R40","unstructured":"L.H. Lim and J. Morton,\n                      Cumulant component analysis: A simultaneous generalization of PCA and ICA\n                      , in Proceedings of Computational Algebraic Statistics, Theories and Applications (CASTA '08), Kyoto University, Kyoto, Japan, Intitute of Statistical Mathematics, Tokyo, Japan, 2008."},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899354688"},{"key":"R42","unstructured":"J. Morton,\n                      Scalable Implicit Symmetric Tensor Approximation\n                      , preprint, Pennsylvania State University, University Park, PA, 2010."},{"key":"R43","unstructured":"J. Morton and L.H. Lim,\n                      Principal Cumulant Components Analysis\n                      , preprint, University of Chicago, Chicago, IL, 2010."},{"key":"R44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jneumeth.2006.11.008"},{"key":"R45","unstructured":"J. Nocedal and S. J. Wright,\n                      Numerical Optimization\n                      , 2nd ed., Springer-Verlag, New York, 2006."},{"key":"R46","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0709146104"},{"key":"R47","unstructured":"I. V. Oseledets,\n                      \n                        Compact Matrix Form of the\n                        d\n                        -dimensional Tensor Decomposition\n                      \n                      , Technical report, Institute of Numerical Mathematics, Russian Academy of Science, Moscow, Russia, 2009."},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1137\/060655894"},{"key":"R49","unstructured":"B. Savas,\n                      Algorithm Package Manual: Best Low Rank Tensor Approximation\n                      , Department of Mathematics, Link\u00f6ping Univeristy, Link\u00f6ping, Sweden, 2008, http:\/\/www.mai.liu.se\/~besav\/soft.html."},{"key":"R50","unstructured":"B. Savas,\n                      Toolbox for Grassmann Manifold Computations\n                      , Department of Mathematics, Link\u00f6ping Univeristy, Link\u00f6ping, Sweden, 2008, http:\/\/www.mai.liu.se\/~besav\/soft.html."},{"key":"R51","unstructured":"L. Simonsson,\n                      Subspace Computations via Matrix Decompositions and Geometric Optimization\n                      , Link\u00f6ping Stud. Sci. Tech., 1052, Link\u00f6ping University, Link\u00f6ping, Sweden, 2007."},{"key":"R52","doi-asserted-by":"crossref","unstructured":"A. Smilde, R. Bro, and P. Geladi,\n                      Multi-way Analysis: Applications in the Chemical Sciences\n                      , John Wiley, West Sussex, England, 2004.","DOI":"10.1002\/0470012110"},{"key":"R53","doi-asserted-by":"publisher","DOI":"10.1007\/BF02289464"},{"key":"R54","first-page":"93","volume":"2","author":"Vasilescu M. A. O.","year":"2003","journal-title":"in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Conference (CVPR '03)"},{"key":"R55","doi-asserted-by":"crossref","unstructured":"T. Yokonuma,\n                      Tensor Spaces and Exterior Algebra\n                      , AMS, Providence, RI, 1992.","DOI":"10.1090\/mmono\/108"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090763172","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:34:29Z","timestamp":1787330069000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090763172"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":55,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090763172"],"URL":"https:\/\/doi.org\/10.1137\/090763172","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}