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Volumn 23, Issue 2, 2013, Pages 1214-1236

Low-rank matrix completion by Riemannian optimization

Author keywords

Differential geometry; Low rank matrices; Matrix completion; Newton; Nonlinear conjugate gradients; Optimization on manifolds; Riemannian manifolds

Indexed keywords

DIFFERENTIAL GEOMETRY; LOW-RANK MATRICES; MATRIX COMPLETION; NONLINEAR CONJUGATE GRADIENT; OPTIMIZATION ON MANIFOLDS; RIEMANNIAN MANIFOLD;

EID: 84880713057     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/110845768     Document Type: Article
Times cited : (528)

References (53)
  • 3
    • 84861639358 scopus 로고    scopus 로고
    • Projection-like retractions on matrix manifolds
    • P.-A. Absil and J. Malick, Projection-like retractions on matrix manifolds, SIAM J. Optim., 22 (2012), pp. 135-158.
    • (2012) SIAM J. Optim. , vol.22 , pp. 135-158
    • Absil, P.-A.1    Malick, J.2
  • 4
    • 0345447647 scopus 로고    scopus 로고
    • Newton's method on Riemannian manifolds and a geometric model for the human spine
    • R. L. Adler, J.-P. Dedieu, J. Y. Margulies, M. Martens, and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal., 22 (2002), pp. 359-390.
    • (2002) IMA J. Numer. Anal. , vol.22 , pp. 359-390
    • Adler, R.L.1    Dedieu, J.-P.2    Margulies, J.Y.3    Martens, M.4    Shub, M.5
  • 10
    • 0003329577 scopus 로고
    • Determinantal rings
    • Springer-Verlag, Berlin
    • W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math. 1327, Springer-Verlag, Berlin, 1988.
    • (1988) Lecture Notes in Math. , vol.1327
    • Bruns, W.1    Vetter, U.2
  • 11
    • 1542337156 scopus 로고    scopus 로고
    • A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization
    • S. Burer and R. D. C. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Math. Program., 95 (2003), pp. 329-357.
    • (2003) Math. Program. , vol.95 , pp. 329-357
    • Burer, S.1    Monteiro, R.D.C.2
  • 12
    • 21644465788 scopus 로고    scopus 로고
    • Local minima and convergence in low-rank semidefinite programming
    • S. Burer and R. D. C. Monteiro, Local minima and convergence in low-rank semidefinite programming, Math. Program., 103 (2005), pp. 427-444.
    • (2005) Math. Program. , vol.103 , pp. 427-444
    • Burer, S.1    Monteiro, R.D.C.2
  • 13
    • 77951291046 scopus 로고    scopus 로고
    • A singular value thresholding algorithm for matrix completion
    • J.-F. Cai, E. J. Candès, and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM J. Optim., 20 (2010), pp. 1956-1982.
    • (2010) SIAM J. Optim. , vol.20 , pp. 1956-1982
    • Cai, J.-F.1    Candès, E.J.2    Shen, Z.3
  • 14
  • 15
    • 77951528523 scopus 로고    scopus 로고
    • The power of convex relaxation: Near-optimal matrix completion
    • E. J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion, IEEE Trans. Inform. Theory, 56 (2009), pp. 2053-2080.
    • (2009) IEEE Trans. Inform. Theory , vol.56 , pp. 2053-2080
    • Candès, E.J.1    Tao, T.2
  • 16
    • 71049116435 scopus 로고    scopus 로고
    • Exact matrix completion via convex optimization
    • E. Candès and B. Recht, Exact matrix completion via convex optimization, Found. Comput. Math., 9 (2009), pp. 717-772.
    • (2009) Found. Comput. Math. , vol.9 , pp. 717-772
    • Candès, E.1    Recht, B.2
  • 17
    • 84872117100 scopus 로고    scopus 로고
    • Iterative methods for low rank approximation of graph similarity matrices
    • T. P. Cason, P.-A. Absil, and P. Van Dooren, Iterative methods for low rank approximation of graph similarity matrices, Linear Algebra Appl., 438 (2013), pp. 1863-1882.
    • (2013) Linear Algebra Appl. , vol.438 , pp. 1863-1882
    • Cason, T.P.1    Absil, P.-A.2    Van Dooren, P.3
  • 18
    • 79959203326 scopus 로고    scopus 로고
    • Subspace evolution and transfer (SET) for low-rank matrix completion
    • W. Dai, O. Milenkovic, and E. Kerman, Subspace evolution and transfer (SET) for low-rank matrix completion, IEEE Trans. Signal Process., 59 (2011), pp. 3120-3132.
    • (2011) IEEE Trans. Signal Process. , vol.59 , pp. 3120-3132
    • Dai, W.1    Milenkovic, O.2    Kerman, E.3
  • 20
    • 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 (1999), pp. 303-353.
    • (1999) SIAM J. Matrix Anal. Appl. , vol.20 , pp. 303-353
    • Edelman, A.1    Arias, T.A.2    Smith, S.T.3
  • 21
    • 79952483985 scopus 로고    scopus 로고
    • Convergence of fixed point continuation algorithms for matrix rank minimization
    • D. Goldfarb and S. Ma, Convergence of fixed point continuation algorithms for matrix rank minimization, Found. Comput. Math., 11 (2011), pp. 183-210.
    • (2011) Found. Comput. Math. , vol.11 , pp. 183-210
    • Goldfarb, D.1    Ma, S.2
  • 22
    • 0004236492 scopus 로고    scopus 로고
    • 3rd ed., Johns Hopkins Stud. Math. Sci., Baltimore, MD
    • G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins Stud. Math. Sci., Baltimore, MD, 1996.
    • (1996) Matrix Computations
    • Golub, G.H.1    Van Loan, C.F.2
  • 23
    • 0000803444 scopus 로고
    • The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate
    • G. H. Golub and V. Pereyra, The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate, SIAM J. Numer. Anal., 10 (1973), pp. 413-432.
    • (1973) SIAM J. Numer. Anal. , vol.10 , pp. 413-432
    • Golub, G.H.1    Pereyra, V.2
  • 24
    • 84857789592 scopus 로고    scopus 로고
    • On the low-rank approximation by the pivoted Cholesky decomposition
    • H. Harbrecht, M. Peters, and R. Schneider, On the low-rank approximation by the pivoted Cholesky decomposition, Appl. Numer. Math., 62 (2012), pp. 428-440.
    • (2012) Appl. Numer. Math. , vol.62 , pp. 428-440
    • Harbrecht, H.1    Peters, M.2    Schneider, R.3
  • 26
    • 77956086700 scopus 로고    scopus 로고
    • Low-rank optimization on the cone of positive semidefinite matrices
    • M. Journée, F. Bach, P.-A. Absil, and R. Sepulchre, Low-rank optimization on the cone of positive semidefinite matrices, SIAM J. Optim., 20 (2010), pp. 2327-2351.
    • (2010) SIAM J. Optim. , vol.20 , pp. 2327-2351
    • Journée, M.1    Bach, F.2    Absil, P.-A.3    Sepulchre, R.4
  • 28
    • 77956909553 scopus 로고    scopus 로고
    • Matrix completion from noisy entries
    • R. Keshavan, A. Montanari, and S. Oh, Matrix completion from noisy entries, J. Mach. Learn. Res., 11 (2010), pp. 2057-2078.
    • (2010) J. Mach. Learn. Res. , vol.11 , pp. 2057-2078
    • Keshavan, R.1    Montanari, A.2    Oh, S.3
  • 29
    • 42649097264 scopus 로고    scopus 로고
    • Dynamical low-rank approximation
    • O. Koch and C. Lubich, Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434-454.
    • (2007) SIAM J. Matrix Anal. Appl. , vol.29 , pp. 434-454
    • Koch, O.1    Lubich, C.2
  • 30
    • 77956045829 scopus 로고    scopus 로고
    • Krylov subspace methods for linear systems with tensor product structure
    • D. Kressner and C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1688-1714.
    • (2010) SIAM J. Matrix Anal. Appl. , vol.31 , pp. 1688-1714
    • Kressner, D.1    Tobler, C.2
  • 32
    • 24344485900 scopus 로고    scopus 로고
    • Introduction to smooth manifolds
    • Springer-Verlag, New York
    • J. M. Lee, Introduction to smooth manifolds, Grad. Texts in Math. 218, Springer-Verlag, New York, 2003.
    • (2003) Grad. Texts in Math. , vol.218
    • Lee, J.M.1
  • 33
    • 77955747588 scopus 로고    scopus 로고
    • ADMiRA: Atomic decomposition for minimum rank approximation
    • K. Lee and Y. Bresler, ADMiRA: Atomic decomposition for minimum rank approximation, IEEE Trans. Inform. Theory, 56 (2010), pp. 4402-4416.
    • (2010) IEEE Trans. Inform. Theory , vol.56 , pp. 4402-4416
    • Lee, K.1    Bresler, Y.2
  • 34
    • 48549094732 scopus 로고    scopus 로고
    • Alternating projections on manifolds
    • A. S. Lewis and J. Malick, Alternating projections on manifolds, Math. Oper. Res., 33 (2008), pp. 216-234.
    • (2008) Math. Oper. Res. , vol.33 , pp. 216-234
    • Lewis, A.S.1    Malick, J.2
  • 36
    • 84862272060 scopus 로고    scopus 로고
    • An implementable proximal point algorithmic framework for nuclear norm minimization
    • Y.-J. Liu, D. Sun, and K.-C. Toh, An implementable proximal point algorithmic framework for nuclear norm minimization, Math. Program., 133 (2012), pp. 399-436.
    • (2012) Math. Program. , vol.133 , pp. 399-436
    • Liu, Y.-J.1    Sun, D.2    Toh, K.-C.3
  • 37
    • 79957957723 scopus 로고    scopus 로고
    • Fixed point and Bregman iterative methods for matrix rank minimization
    • S. Ma, D. Goldfarb, and L. Chen, Fixed point and Bregman iterative methods for matrix rank minimization, Math. Program., 128 (2011), pp. 321-353.
    • (2011) Math. Program. , vol.128 , pp. 321-353
    • Ma, S.1    Goldfarb, D.2    Chen, L.3
  • 40
    • 79952746338 scopus 로고    scopus 로고
    • Regression on fixed-rank positive semidefinite matrices: A Riemannian approach
    • G. Meyer, S. Bonnabel, and R. Sepulchre, Regression on fixed-rank positive semidefinite matrices: A Riemannian approach, J. Mach. Learn. Res., 12 (2011), pp. 593-625.
    • (2011) J. Mach. Learn. Res. , vol.12 , pp. 593-625
    • Meyer, G.1    Bonnabel, S.2    Sepulchre, R.3
  • 45
    • 33748923977 scopus 로고    scopus 로고
    • A Newton-like method for solving rank constrained linear matrix inequalities
    • R. Orsi, U. Helmke, and J. B. Moore, A Newton-like method for solving rank constrained linear matrix inequalities, Automatica, 42 (2006), pp. 1875-1882.
    • (2006) Automatica , vol.42 , pp. 1875-1882
    • Orsi, R.1    Helmke, U.2    Moore, J.B.3
  • 47
    • 85162007899 scopus 로고    scopus 로고
    • Online learning in the manifold of low-rank matrices
    • J. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. S. Zemel, and A. Culotta, eds.
    • U. Shalit, D. Weinshall, and G. Chechik, Online learning in the manifold of low-rank matrices, in Advances in Neural Inf. Process. Syst. 23, J. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. S. Zemel, and A. Culotta, eds., 2010, pp. 2128-2136.
    • (2010) Advances in Neural Inf. Process. Syst. , vol.23 , pp. 2128-2136
    • Shalit, U.1    Weinshall, D.2    Chechik, G.3
  • 48
    • 84857807897 scopus 로고    scopus 로고
    • Online learning in the embedded manifold of low-rank matrices
    • U. Shalit, D. Weinshall, and G. Chechik, Online learning in the embedded manifold of low-rank matrices, J. Mach. Learn. Res., 13 (2012), pp. 429-458.
    • (2012) J. Mach. Learn. Res. , vol.13 , pp. 429-458
    • Shalit, U.1    Weinshall, D.2    Chechik, G.3
  • 49
    • 0000188055 scopus 로고
    • Optimization techniques on Riemannian manifold
    • Hamiltonian and Gradient Flows, AMS, Providence, RI
    • S. T. Smith, Optimization techniques on Riemannian manifold, in Hamiltonian and Gradient Flows, Algorithms and Control, vol. 3, AMS, Providence, RI, 1994, pp. 113-136.
    • (1994) Algorithms and Control , vol.3 , pp. 113-136
    • Smith, S.T.1
  • 50
    • 78049448383 scopus 로고    scopus 로고
    • An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems
    • K. Toh and S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems, Pacific J. Optimization, 6 (2010), pp. 615-640.
    • (2010) Pacific J. Optimization , vol.6 , pp. 615-640
    • Toh, K.1    Yun, S.2
  • 51
    • 79251469735 scopus 로고    scopus 로고
    • A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations
    • B. Vandereycken and S. Vandewalle, A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2553-2579.
    • (2010) SIAM J. Matrix Anal. Appl. , vol.31 , pp. 2553-2579
    • Vandereycken, B.1    Vandewalle, S.2
  • 53
    • 84869205493 scopus 로고    scopus 로고
    • Solving a low-rank factorization model for matrix completion by a non-linear successive over-relaxation algorithm
    • Z.Wen, W. Yin, and Yin Zhang, Solving a low-rank factorization model for matrix completion by a non-linear successive over-relaxation algorithm, Math. Program. Comput., 4 (2012), pp. 333-361.
    • (2012) Math. Program. Comput. , vol.4 , pp. 333-361
    • Wen, Z.1    Yin, W.2    Zhang, Y.3


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