메뉴 건너뛰기




Volumn 33, Issue 3, 2012, Pages 806-821

The best rank-1 approximation of a symmetric tensor and related spherical optimization problems

Author keywords

Power algorithm; Symmetric tensor; The best rank 1 approximation; The best symmetric rank 1 approximation

Indexed keywords

APPROXIMATION RATIOS; HOMOGENEOUS POLYNOMIALS; LOWER BOUNDS; NUMERICAL RESULTS; OPTIMIZATION PROBLEMS; ORDER POLYNOMIALS; THE BEST RANK-1 APPROXIMATION; THE BEST SYMMETRIC RANK-1 APPROXIMATION;

EID: 84867305742     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/110835335     Document Type: Article
Times cited : (104)

References (36)
  • 3
    • 70450271025 scopus 로고    scopus 로고
    • A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the orientation distribution function in diffusion MRI
    • New York
    • A. Ghosh, E. Tsigaridas, M. Descoteaux, P. Comon, B. Mourrain, and R. Deriche, A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the orientation distribution function in diffusion MRI, Computational Diffusion MRI Workshop, New York, 2008.
    • (2008) Computational Diffusion MRI Workshop
    • Ghosh, A.1    Tsigaridas, E.2    Descoteaux, M.3    Comon, P.4    Mourrain, B.5    Deriche, R.6
  • 5
    • 77958088475 scopus 로고    scopus 로고
    • Approximation algorithms for homogeneous polynomial optimization with quadratic constraints
    • S. He, Z. Li, and S. Zhang, Approximation algorithms for homogeneous polynomial optimization with quadratic constraints, Math. Program., 125 (2011), pp. 353-383.
    • (2011) Math. Program. , vol.125 , pp. 353-383
    • He, S.1    Li, Z.2    Zhang, S.3
  • 6
    • 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
  • 7
    • 84856302455 scopus 로고    scopus 로고
    • Shifted power method for computing tensor eigenpairs
    • T.G. Kolda and J.R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1095-1124.
    • (2011) SIAM J. Matrix Anal. Appl. , vol.32 , pp. 1095-1124
    • Kolda, T.G.1    Mayo, J.R.2
  • 8
    • 0004480637 scopus 로고
    • Real and functional analysis
    • Springer-Verlag, New York
    • S. Lang, Real and functional analysis, Grad. Texts in Math. 142, Springer-Verlag, New York, 1993.
    • (1993) Grad. Texts in Math. , vol.142
    • Lang, S.1
  • 9
    • 0035238864 scopus 로고    scopus 로고
    • Global optimization with polynomials and the problem of moments
    • J.B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796-817.
    • (2001) SIAM J. Optim. , vol.11 , pp. 796-817
    • Lasserre, J.B.1
  • 10
    • 0036034519 scopus 로고    scopus 로고
    • Polynomials nonnegative on a grid and discrete representations
    • J.B. Lasserre, Polynomials nonnegative on a grid and discrete representations, Trans. Amer. Math. Soc., 354 (2001), pp. 631-649.
    • (2001) Trans. Amer. Math. Soc. , vol.354 , pp. 631-649
    • Lasserre, J.B.1
  • 11
    • 2642555525 scopus 로고    scopus 로고
    • 3) approximation in multilinear algebra
    • 3) approximation in multilinear algebra, J. Chemometrics, 18 (2004), pp. 2-11.
    • (2004) J. Chemometrics , Issue.18 , pp. 2-11
    • De Lathauwer, L.1
  • 13
    • 67349148256 scopus 로고    scopus 로고
    • Sums of squares, moment matrices and optimization over polynomials
    • M. Putinar and S. Sullivant, eds., Springer, New York
    • M. Laurent, Sums of squares, moment matrices and optimization over polynomials, in Emerging Applications of Algebraic Geometry, IMA Volumes in Mathematics and its Applications 149, M. Putinar and S. Sullivant, eds., Springer, New York, 2009, pp. 157-270.
    • (2009) Emerging Applications of Algebraic Geometry, IMA Volumes in Mathematics and Its Applications , vol.149 , pp. 157-270
    • Laurent, M.1
  • 14
    • 33144459397 scopus 로고    scopus 로고
    • A new gradient search interpretation of super exponential algorithm
    • X.L. Li, A new gradient search interpretation of super exponential algorithm, IEEE Signal Process. Lett., 13 (2006), pp. 173-176.
    • (2006) IEEE Signal Process. Lett. , vol.13 , pp. 173-176
    • Li, X.L.1
  • 15
    • 77951288110 scopus 로고    scopus 로고
    • A semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints
    • Z.Q. Luo and S. Zhang, A semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints, SIAM J. Optim., 20 (2010), pp. 1716-1736.
    • (2010) SIAM J. Optim. , vol.20 , pp. 1716-1736
    • Luo, Z.Q.1    Zhang, S.2
  • 16
    • 84995186518 scopus 로고
    • Portfolio selection
    • H.M. Markowitz, Portfolio selection, J. Finance, 7 (1952), pp. 77-91.
    • (1952) J. Finance , vol.7 , pp. 77-91
    • Markowitz, H.M.1
  • 21
    • 34548409985 scopus 로고    scopus 로고
    • On the best rank-1 approximation to higher-order symmetric tensor
    • G. Ni and Y. Wang, On the best rank-1 approximation to higher-order symmetric tensor, Math. Comput. Modelling, 46 (2007), pp. 1345-1352.
    • (2007) Math. Comput. Modelling , vol.46 , pp. 1345-1352
    • Ni, G.1    Wang, Y.2
  • 22
    • 51749116732 scopus 로고    scopus 로고
    • An eigenvalue method for testing positive definiteness of a multivariate form
    • Q. Ni, L. Qi, and F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Control, 53 (2008), pp. 1096-1107.
    • (2008) IEEE Trans. Automat. Control , vol.53 , pp. 1096-1107
    • Ni, Q.1    Qi, L.2    Wang, F.3
  • 23
    • 84858438014 scopus 로고    scopus 로고
    • Sum of squares methods for minimizing polynomial forms over spheres and hypersurfaces
    • J. Nie, Sum of squares methods for minimizing polynomial forms over spheres and hypersurfaces, Frontiers of Mathematics in China, 7 (2012), pp. 321-346.
    • (2012) Frontiers of Mathematics in China , vol.7 , pp. 321-346
    • Nie, J.1
  • 24
    • 1542266056 scopus 로고    scopus 로고
    • Semidefinite programming relaxations for semialgebraic problems
    • P.A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Program., 96 (2003), pp. 293-320.
    • (2003) Math. Program. , vol.96 , pp. 293-320
    • Parrilo, P.A.1
  • 25
    • 14644427109 scopus 로고    scopus 로고
    • Extrema of a real polynomial
    • L. Qi, Extrema of a real polynomial, J. Global Optim., 30 (2004), pp. 405-433.
    • (2004) J. Global Optim. , vol.30 , pp. 405-433
    • Qi, L.1
  • 26
    • 27744582534 scopus 로고    scopus 로고
    • Eigenvalues of a real supersymmetric tensor
    • L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), pp. 1302-1324.
    • (2005) J. Symbolic Comput. , vol.40 , pp. 1302-1324
    • Qi, L.1
  • 27
    • 80051526441 scopus 로고    scopus 로고
    • The best rank-one approximation ratio of a tensor space
    • L. Qi, The best rank-one approximation ratio of a tensor space, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 430-442.
    • (2011) SIAM J. Matrix Anal. Appl. , vol.32 , pp. 430-442
    • Qi, L.1
  • 28
    • 14944344547 scopus 로고    scopus 로고
    • Global minimization of normal quadratic polynomials based on global descent directions
    • L. Qi, Z. Wan, and Y.F. Yang, Global minimization of normal quadratic polynomials based on global descent directions, SIAM J. Optim., 15 (2004), pp. 275-302.
    • (2004) SIAM J. Optim. , vol.15 , pp. 275-302
    • Qi, L.1    Wan, Z.2    Yang, Y.F.3
  • 29
    • 58149504770 scopus 로고    scopus 로고
    • Z-eigenvalue methods for a global polynomial optimization problem
    • L. Qi, F. Wang, and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Math. Program., 118 (2009), pp. 301-316.
    • (2009) Math. Program. , vol.118 , pp. 301-316
    • Qi, L.1    Wang, F.2    Wang, Y.3
  • 30
    • 0042525811 scopus 로고    scopus 로고
    • Monotonic convergence of fixed-point algorithms for ICA
    • P.A. Regalia and E. Kofides, Monotonic convergence of fixed-point algorithms for ICA, IEEE Trans. Neural Networks, 14 (2003), pp. 943-949.
    • (2003) IEEE Trans. Neural Networks , vol.14 , pp. 943-949
    • Regalia, P.A.1    Kofides, E.2
  • 31
    • 81155134640 scopus 로고    scopus 로고
    • Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems
    • A. M-C. So, Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems, Math. Program., 129 (2011), pp. 357-382.
    • (2011) Math. Program. , vol.129 , pp. 357-382
    • So, A.M.-C.1
  • 32
    • 40849130164 scopus 로고    scopus 로고
    • On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet forming
    • S. Soare, J.W. Yoon, and O. Cazacu, On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet forming, Internat. J. Plasticity, 24 (2008), pp. 915-944.
    • (2008) Internat. J. Plasticity , vol.24 , pp. 915-944
    • Soare, S.1    Yoon, J.W.2    Cazacu, O.3
  • 34
    • 76749172184 scopus 로고    scopus 로고
    • Robust independent component analysis by iterative maximization of kurtosis contrast with algebraic optimal step size
    • V. Zarzoso and P. Comon, Robust independent component analysis by iterative maximization of kurtosis contrast with algebraic optimal step size, IEEE Trans. Neural Networks, 21 (2010), pp. 534-550.
    • (2010) IEEE Trans. Neural Networks , vol.21 , pp. 534-550
    • Zarzoso, V.1    Comon, P.2
  • 35
    • 0036056838 scopus 로고    scopus 로고
    • Rank-1 approximation of higher-order tensors
    • T. Zhang and G.H. Golub, Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 534-550.
    • (2001) SIAM J. Matrix Anal. Appl. , vol.23 , pp. 534-550
    • Zhang, T.1    Golub, G.H.2
  • 36
    • 84867295918 scopus 로고    scopus 로고
    • The cubic spherical optimization problems
    • X.Z. Zhang, L. Qi, and Y. Ye, The cubic spherical optimization problems, Math. Comput., 81 (2012), pp. 1513-1525.
    • (2012) Math. Comput. , vol.81 , pp. 1513-1525
    • Zhang, X.Z.1    Qi, L.2    Ye, Y.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.