메뉴 건너뛰기




Volumn 32, Issue 6, 2010, Pages 3352-3393

Quasi-Newton methods on Grassmannians and multilinear approximations of tensors

Author keywords

Approximations; Grassmann BFGS; Grassmann limited memory BFGS; Grassmann manifold; Grassmann quasi Newton; Grassmannian; Multilinear rank; Product of Grassmannians; Symmetric multilinear rank; Symmetric tensor; Tensor

Indexed keywords

APPROXIMATION ALGORITHMS; NEWTON-RAPHSON METHOD;

EID: 79251508042     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/090763172     Document Type: Article
Times cited : (116)

References (55)
  • 4
    • 33845525246 scopus 로고    scopus 로고
    • Algorithm 862: MATLAB tensor classes for fast algorithm prototyping
    • B. W. Bader and T. G. Kolda, Algorithm 862: MATLAB tensor classes for fast algorithm prototyping, ACM Trans. Math. Software, 32 (2006), pp. 635-653.
    • (2006) ACM Trans. Math. Software , vol.32 , pp. 635-653
    • Bader, B.W.1    Kolda, T.G.2
  • 6
    • 0028319529 scopus 로고
    • Representations of quasi-Newton matrices and their use in limited memory methods
    • R. H. Byrd, J. Nocedal, and R. B. Schnabel, Representations of quasi-Newton matrices and their use in limited memory methods, Math. Program., 63 (1994), pp. 129-156.
    • (1994) Math. Program. , vol.63 , pp. 129-156
    • Byrd, R.H.1    Nocedal, J.2    Schnabel, R.B.3
  • 7
    • 0039991939 scopus 로고    scopus 로고
    • A modified Cholesky algorithm based on a symmetric indefinite factorization
    • S. H. Cheng and N. J. Higham, A modified Cholesky algorithm based on a symmetric indefinite factorization, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 1097-1110.
    • (1998) SIAM J. Matrix Anal. Appl. , vol.19 , pp. 1097-1110
    • Cheng, S.H.1    Higham, N.J.2
  • 10
    • 0030234706 scopus 로고    scopus 로고
    • Decomposition of quantics in sums of powers of linear forms
    • P. Comon and B. Mourrain, Decomposition of quantics in sums of powers of linear forms, Signal Process., 53 (1996), pp. 93-107.
    • (1996) Signal Process. , vol.53 , pp. 93-107
    • Comon, P.1    Mourrain, B.2
  • 12
    • 79251505752 scopus 로고    scopus 로고
    • Tucker compression, parallel factor analysis and block term decompositions: New results
    • Technical University of Denmark, Copenhagen, Denmark
    • L. De Lathauwer, Tucker compression, parallel factor analysis and block term decompositions: New results, European Meeting on Challenges in Modern Massive Data Sets (EMMDS '09), Technical University of Denmark, Copenhagen, Denmark, 2009.
    • (2009) European Meeting on Challenges in Modern Massive Data Sets (EMMDS '09)
    • De Lathauwer, L.1
  • 13
    • 0003450304 scopus 로고    scopus 로고
    • Ph.D. thesis, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium
    • L. De Lathauwer, Signal Processing Based on Multilinear Algebra, Ph.D. thesis, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, 1997.
    • (1997) Signal Processing Based on Multilinear Algebra
    • De Lathauwer, L.1
  • 14
    • 0033653097 scopus 로고    scopus 로고
    • An introduction to independent component analysis
    • L. De Lathauwer, B. De Moor, and J. Vandewalle, An introduction to independent component analysis, J. Chemometrics, 14 (2000), pp. 123-149.
    • (2000) J. Chemometrics , vol.14 , pp. 123-149
    • De Lathauwer, L.1    De Moor, B.2    Vandewalle, J.3
  • 16
    • 0034144761 scopus 로고    scopus 로고
    • On the best rank-1 and rank- (R1,R2, . . . ,RN) approximation of higher-order tensors
    • L. De Lathauwer, B. De Moor, and J. Vandewalle, On the best rank-1 and rank- (R1,R2, . . . ,RN) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1324-1342.
    • (2000) SIAM J. Matrix Anal. Appl. , vol.21 , pp. 1324-1342
    • De Lathauwer, L.1    De Moor, B.2    Vandewalle, J.3
  • 17
    • 4644268396 scopus 로고    scopus 로고
    • Dimensionality reduction in higher-order signal processing and rank-(r1, r2, . . . , rn) reduction in multilinear algebra
    • L. De Lathauwer and J. Vandewalle, Dimensionality reduction in higher-order signal processing and rank-(r1, r2, . . . , rn) reduction in multilinear algebra, Linear Algebra Appl., 391 (2004), pp. 31-55.
    • (2004) Linear Algebra Appl. , vol.391 , pp. 31-55
    • De Lathauwer, L.1    Vandewalle, J.2
  • 19
    • 55349142218 scopus 로고    scopus 로고
    • Tensor rank and the ill-posedness of the best low-rank approximation problem
    • V. De Silva and L.-H. Lim, Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1084-1127.
    • (2008) SIAM J. Matrix Anal. Appl. , vol.30 , pp. 1084-1127
    • De Silva, V.1    Lim, L.-H.2
  • 20
    • 0002663672 scopus 로고
    • Quasi-Newton methods, motivation and theory
    • J. E. Dennis and J. Moré, Quasi-Newton methods, motivation and theory, SIAM Rev., 19 (1977), pp. 46-89.
    • (1977) SIAM Rev. , vol.19 , pp. 46-89
    • Dennis, J.E.1    Moré, J.2
  • 21
    • 0000217813 scopus 로고
    • A new derivation of symmetric positive definite secant updates
    • J. E. Dennis and R. B. Schnabel, A new derivation of symmetric positive definite secant updates, Nonlinear Program., 4 (1981), pp. 167-199.
    • (1981) Nonlinear Program. , vol.4 , pp. 167-199
    • Dennis, J.E.1    Schnabel, R.B.2
  • 23
    • 0032216898 scopus 로고    scopus 로고
    • The geometry of algorithms with orthogonality constraints
    • A. Edelman, T. A. Arias, and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 303-353.
    • (1998) SIAM J. Matrix Anal. Appl. , vol.20 , pp. 303-353
    • Edelman, A.1    Arias, T.A.2    Smith, S.T.3
  • 24
    • 68149125583 scopus 로고    scopus 로고
    • A Newton-Grassmann method for computing the best multilinear rank- (r1, r2, r3) approximation of a tensor
    • L. Eldén and B. Savas, A Newton-Grassmann method for computing the best multilinear rank- (r1, r2, r3) approximation of a tensor, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 248-271.
    • (2009) SIAM J. Matrix Anal. Appl. , vol.31 , pp. 248-271
    • Eldén, L.1    Savas, B.2
  • 26
    • 0020145445 scopus 로고
    • Minimizing a differentiable function over a differential manifold
    • D. Gabay, Minimizing a differentiable function over a differential manifold, J. Optim. Theory Appl., 37 (1982), pp. 177-219.
    • (1982) J. Optim. Theory Appl. , vol.37 , pp. 177-219
    • Gabay, D.1
  • 27
    • 84954050397 scopus 로고
    • Factorized variable metric methods for unconstrained optimization
    • D. Goldfarb, Factorized variable metric methods for unconstrained optimization, Math. Comp., 30 (1976), pp. 796-811.
    • (1976) Math. Comp. , vol.30 , pp. 796-811
    • Goldfarb, D.1
  • 28
    • 0004236492 scopus 로고    scopus 로고
    • 3rd ed. Johns Hopkins University Press, Baltimore, MD
    • G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
    • (1996) Matrix Computations
    • Golub, G.H.1    Van Loan, C.F.2
  • 29
    • 0004016604 scopus 로고
    • 2nd ed. Springer-Verlag, New York, NY
    • W. Greub, Multilinear Algebra, 2nd ed., Springer-Verlag, New York, NY, 1978.
    • (1978) Multilinear Algebra
    • Greub, W.1
  • 30
    • 85162707890 scopus 로고
    • The expression of a tensor or a polyadic as a sum of products
    • F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of products, J. Math. Phys., 6 (1927), pp. 164-189.
    • (1927) J. Math. Phys. , vol.6 , pp. 164-189
    • Hitchcock, F.L.1
  • 31
    • 67651114913 scopus 로고    scopus 로고
    • Dimensionality reduction for higher-order tensors: Algorithms and applications
    • M. Ishteva, L. De Lathauwer, P.-A. Absil, and S. Van Huffel, Dimensionality reduction for higher-order tensors: Algorithms and applications, Int. J. Pure Appl. Math., 42 (2008), pp. 337-343.
    • (2008) Int. J. Pure Appl. Math. , vol.42 , pp. 337-343
    • Ishteva, M.1    De Lathauwer, L.2    Absil, P.-A.3    Van Huffel, S.4
  • 33
    • 67349104920 scopus 로고    scopus 로고
    • Differential-geometric Newton algorithm for the best rank-(r1, r2, r3) approximation of tensors
    • M. Ishteva, L. De Lathauwer, P.-A. Absil, and S. Van Huffel, Differential-geometric Newton algorithm for the best rank-(r1, r2, r3) approximation of tensors, Numer. Algorithms, 51 (2009), pp. 179-194.
    • (2009) Numer. Algorithms , vol.51 , pp. 179-194
    • Ishteva, M.1    De Lathauwer, L.2    Absil, P.-A.3    Van Huffel, S.4
  • 34
    • 34447513756 scopus 로고    scopus 로고
    • Low rank Tucker-type tensor approximation to classical potentials
    • B. N. Khoromskij and V. Khoromskaia, Low rank Tucker-type tensor approximation to classical potentials, Cent. Eur. J. Math., 5 (2007), pp. 523-550.
    • (2007) Cent. Eur. J. Math. , vol.5 , pp. 523-550
    • Khoromskij, B.N.1    Khoromskaia, V.2
  • 35
    • 84897586218 scopus 로고    scopus 로고
    • Multigrid accelerated tensor approximation of function related multidimensional arrays
    • B. N. Khoromskij and V. Khoromskaia, Multigrid accelerated tensor approximation of function related multidimensional arrays, SIAM J. Sci. Comput., 31 (2009), pp. 3002-3026.
    • (2009) SIAM J. Sci. Comput. , vol.31 , pp. 3002-3026
    • Khoromskij, B.N.1    Khoromskaia, V.2
  • 36
    • 0036018642 scopus 로고    scopus 로고
    • On the best rank-1 approximation of higher-order supersymmetric tensors
    • E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 863-884.
    • (2002) SIAM J. Matrix Anal. Appl. , vol.23 , pp. 863-884
    • Kofidis, E.1    Regalia, P.A.2
  • 37
    • 68649096448 scopus 로고    scopus 로고
    • Tensor decompositions and applications
    • T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500.
    • (2009) SIAM Rev. , vol.51 , pp. 455-500
    • Kolda, T.G.1    Bader, B.W.2
  • 38
    • 14844289898 scopus 로고    scopus 로고
    • 3rd ed. Springer-Verlag, New York
    • S. Lang, Algebra, rev. 3rd ed., Springer-Verlag, New York, 2002.
    • (2002) Algebra, Rev.
    • Lang, S.1
  • 40
    • 79251515968 scopus 로고    scopus 로고
    • Cumulant component analysis: A simultaneous generalization of PCA and ICA
    • Kyoto University, Kyoto, Japan, Intitute of Statistical Mathematics, Tokyo, Japan
    • L.-H. Lim and J. Morton, Cumulant component analysis: A simultaneous generalization of PCA and ICA, in Proceedings of Computational Algebraic Statistics, Theories and Applications (CASTA '08), Kyoto University, Kyoto, Japan, Intitute of Statistical Mathematics, Tokyo, Japan, 2008.
    • (2008) Proceedings of Computational Algebraic Statistics, Theories and Applications (CASTA '08)
    • Lim, L.-H.1    Morton, J.2
  • 41
    • 0036018640 scopus 로고    scopus 로고
    • Adaptive eigenvalue computations using Newton's method on the Grassmann manifold
    • E. Lundström and L. Eldén, Adaptive eigenvalue computations using Newton's method on the Grassmann manifold, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 819-839.
    • (2002) SIAM J. Matrix Anal. Appl. , vol.23 , pp. 819-839
    • Lundström, E.1    Eldén, L.2
  • 44
    • 33947514540 scopus 로고    scopus 로고
    • ERPWAVELAB: A toolbox for multi-channel analysis of time-frequency transformed event related potentials
    • M. Mørup, L. K. Hansen, and S. M. Arnfred, ERPWAVELAB: A toolbox for multi-channel analysis of time-frequency transformed event related potentials, J. Neurosci. Methods, 161 (2007), pp. 361-368.
    • (2007) J. Neurosci. Methods , vol.161 , pp. 361-368
    • Mørup, M.1    Hansen, L.K.2    Arnfred, S.M.3
  • 46
    • 36749028922 scopus 로고    scopus 로고
    • A tensor higher-order singular value decomposition for integrative analysis of DNA microarray data from different studies
    • L. Omberg, G. H. Golub, and O. Alter, A tensor higher-order singular value decomposition for integrative analysis of DNA microarray data from different studies, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 18371-18376.
    • (2007) Proc. Natl. Acad. Sci. USA , vol.104 , pp. 18371-18376
    • Omberg, L.1    Golub, G.H.2    Alter, O.3
  • 48
    • 51849118006 scopus 로고    scopus 로고
    • Tucker dimensionality reduction of three-dimensional arrays in linear time
    • I. V. Oseledets, D. V. Savostianov, and E. E. Tyrtyshnikov, Tucker dimensionality reduction of three-dimensional arrays in linear time, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 939-956.
    • (2008) SIAM J. Matrix Anal. Appl. , vol.30 , pp. 939-956
    • Oseledets, I.V.1    Savostianov, D.V.2    Tyrtyshnikov, E.E.3
  • 51
    • 79251513820 scopus 로고    scopus 로고
    • Subspace Computations via Matrix Decompositions and Geometric Optimization, Link
    • Linköping University, Link?oping, Sweden
    • L. Simonsson, Subspace Computations via Matrix Decompositions and Geometric Optimization, Link?oping Stud. Sci. Tech., 1052, Linköping University, Link?oping, Sweden, 2007.
    • (2007) Oping Stud. Sci. Tech. , vol.1052
    • Simonsson, L.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.