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Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries

Bibliographic reference Absil, Pierre-Antoine ; Amodei, Luca ; Meyer, Gilles. Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries. In: Computational Statistics, Vol. 29, no. 3-4, p. 569-590 (2013)
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  1. Abraham Ralph, Marsden Jerrold E., Ratiu Tudor, Manifolds, Tensor Analysis, and Applications, ISBN:9781461269908, 10.1007/978-1-4612-1029-0
  2. Absil PA, Mahony R, Sepulchre R (2008) Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton.
  3. Absil P.-A., Baker C.G., Gallivan K.A., Trust-Region Methods on Riemannian Manifolds, 10.1007/s10208-005-0179-9
  4. Adler R. L., Newton's method on Riemannian manifolds and a geometric model for the human spine, 10.1093/imanum/22.3.359
  5. Amodei L, Dedieu JP, Yakoubsohn JC (2009) A dynamical approach to low-rank approximation of a matrix. In: Communication at the 14th Belgian-French-German conference on optimization, 14–18 September 2009
  6. Boumal N, Absil PA (2011) RTRMC: a Riemannian trust-region method for low-rank matrix completion. In: Shawe-Taylor J, Zemel R, Bartlett P, Pereira F, Weinberger K (eds) Advances in neural information processing systems 24 (NIPS), pp 406–414,
  7. Boumal N, Mishra B (2013) The Manopt toolbox. , version 1.0.1
  8. Dai Wei, Milenkovic Olgica, Kerman Ely, Subspace Evolution and Transfer (SET) for Low-Rank Matrix Completion, 10.1109/tsp.2011.2144977
  9. Dai Wei, Kerman Ely, Milenkovic Olgica, A Geometric Approach to Low-Rank Matrix Completion, 10.1109/tit.2011.2171521
  10. Dennis JE Jr, Schnabel RB (1983) Numerical methods for unconstrained optimization and nonlinear equations. Prentice Hall series in computational mathematics. Prentice Hall Inc., Englewood Cliffs
  11. do Carmo MP (1992) Riemannian geometry. Mathematics: theory & applications, Birkhäuser Boston Inc., Boston, translated from the second Portuguese edition by Francis Flaherty
  12. Edelman Alan, Arias Tomás A., Smith Steven T., The Geometry of Algorithms with Orthogonality Constraints, 10.1137/s0895479895290954
  13. Griewank A, Reddien GW (1985) The approximation of simple singularities. In: Numerical boundary value ODEs (Vancouver, B.C., 1984), Progr. Sci. Comput., vol 5, Birkhäuser, pp 245–259
  14. Helmke Uwe, Moore John B., Optimization and Dynamical Systems, ISBN:9781447134695, 10.1007/978-1-4471-3467-1
  15. Helmke Uwe, Shayman Mark A., Critical points of matrix least squares distance functions, 10.1016/0024-3795(93)00070-g
  16. Keshavan RH, Montanari A, Oh S (2010) Matrix completion from noisy entries. arXiv:0906.2027v2
  17. Meyer G (2011) Geometric optimization algorithms for linear regression on fixed-rank matrices. PhD thesis, University of Liège
  18. Mishra B, Apuroop KA, Sepulchre R (2012a) A Riemannian geometry for low-rank matrix completion. arXiv:1211.1550
  19. Mishra B, Meyer G, Bach F, Sepulchre R (2011a) Low-rank optimization with trace norm penalty. arXiv:1112.2318v1
  20. Mishra B, Meyer G, Bonnabel S, Sepulchre R (2012b) Fixed-rank matrix factorizations and Riemannian low-rank optimization. arXiv:1209.0430v1
  21. Mishra B, Meyer G, Sepulchre R (2011b) Low-rank optimization for distance matrix completion. In: Decision and control and European control conference (CDC-ECC), 2011 50th IEEE conference on, pp 4455–4460
  22. O'Neill Barrett, The fundamental equations of a submersion., 10.1307/mmj/1028999604
  23. O’Neill B (1983) Semi-Riemannian geometry, pure and applied mathematics, vol 103. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York
  24. Simonsson Lennart, Eldén Lars, Grassmann algorithms for low rank approximation of matrices with missing values, 10.1007/s10543-010-0253-9
  25. Smith ST (1994) Optimization techniques on Riemannian manifolds. In: Bloch A (ed) Hamiltonian and gradient flows, algorithms and control, Fields Inst. Commun., vol 3. Am. Math. Soc. Providence, RI, pp 113–136
  26. Vandereycken Bart, Low-Rank Matrix Completion by Riemannian Optimization, 10.1137/110845768