Low-rank optimization for distance matrix completion

Proceedings of the IEEE Conference on Decision and Control

Volumn , Issue , 2011, Pages 4455-4460

  Mishra, B ^a   Meyer, G ^a   Sepulchre, R ^a  

^a UNIVERSITY OF LIÈGE   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

BENCHMARKING; EMBEDDINGS;

DATA EMBEDDING; DISTANCE MATRICES; GLOBAL SOLUTIONS; HIGH-DIMENSIONAL PROBLEMS; NUMERICAL EXPERIMENTS; OPTIMIZATION PROBLEMS; POSITIVE SEMIDEFINITE MATRICES; RANK DISTANCE;

MATRIX ALGEBRA;

EID: 84860686762     PISSN: 07431546     EISSN: 25762370     Source Type: Conference Proceeding    
DOI: 10.1109/CDC.2011.6160810     Document Type: Conference Paper

References (31)

1. E. J. Candes and B. Recht, "Exact matrix completion via convex optimization," Foundations of Computational Mathematics, vol. 9, no. 6, pp. 717-772, Dec. 2009.
2. "The Netflix prize." [Online]. Available: http://www. netflixprize.com/
3. A. Morrison, G. Ross, and M. Chalmers, "Fast multidimensional scaling through sampling, springs and interpolation," Information Visualization, vol. 2, no. 1, pp. 68-77, 2003.
4. I. Borg and P. J. Groenen, Modern multidimensional scaling : theory and applications, ser. Springer series in Statistics. Springer, 1997.
5. J. More and Z. Wu, "Distance geometry optimization for protein structures," Journal of Global Optimization, vol. 15, no. 3, pp. 219-234, Oct. 1999.
6. W. Glunt, T. Hayden, and M. Rayden, "Molecular confirmations from distance matrices," Journal of Computational Chemistry, vol. 14, pp. 114-120, 1993.
7. J. Dattorro, Convex optimization & Euclidean distance geometry. PO Box 12, Palo Alto, CA 94302: Meboo Publishing USA, 2005.
8. A. Y. Alfakih, "On the uniqueness of Euclidean distance matrix completions: the case of points in general position," Linear Algebra Applications, vol. 397, pp. 265-277, 2005.
9. J. D. Leeuw, "Multidimensional scaling," Department of Statistics, UCLA, Tech. Rep., 2001. [Online]. Available: http://preprints.stat. ucla.edu/274/274.pdf
10. A. Kearsley, R. Tapia, and M. Trosset, "The solution of the metric stress and sstress problems in multidimensional scaling by Newton's method," Computational Statistics, vol. 13, no. 3, pp. 369-396, 1998.
11. M. Laurent, "A connection between positive semidefinite and Euclidean distance matrix completion problems," Linear Algebra and its Applications, vol. 273, pp. 9-22, 1998.
12. H.-X. Huang, Z.-A. Liang, and P. M. Pardalos, "Some properties for the Euclidean distance matrix and positive semidefinite matrix completion problems," Journal of Global Optimization, vol. 25, pp. 3-21, January 2003.
13. A. Alfakih, A. Khandani, and H. Wolkowicz, "Solving Euclidean distance matrix completion problems via semidefinite programming," Computational Optimization and Applications, vol. 12, no. 1-3, pp. 13-30, Jan. 1999.
14. L. Cayton and S. Dasgupta, "Robust Euclidean embedding," in Proceedings of the 23rd international conference on Machine learning, 2006.
15. S. Boyd and L. Vandenberghe, Convex optimization. Cambridge University Press, March 2004.
16. M. W. Trosset, "Distance matrix completion by numerical optimization," Comput. Optim. Appl., vol. 17, pp. 11-22, October 2000.
17. J. Kruskal, "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis," Psychometrika, vol. 29, no. 1, pp. 1-27, March 1964.
18. A. Buja and D. F. Swayne, "Interactive data visualization with multidimensional scaling," Department of Statistics, Stanford University, Tech. Rep., March 2004. [Online]. Available: http://www-stat.stanford.edu/ $\sim$tibs/sta306b/buja.pdf
19. P. Tarazaga and M. W. Trosset, "An optimization problem on subsets of the symmetric positive-semidefinite matrices," Journal of Optimization Theory and Applications, vol. 79, pp. 513-524, 1993.
20. H. Fang and D. P. Oleary, "Euclidean distance matrix completion problems," June 2010.
21. P. A. Absil, M. Ishteva, L. De Lathauwer, and S. Van Huffel, "A geometric Newton method for Oja's vector field," Neural Computation, vol. 21, no. 5, pp. 1415-1433, May 2009.
22. D. I. Chu, H. C. Brown, and M. T. Chu, "On least squares Euclidean distance matrix approximation and completion," Department of Mathematics, North Carolina State University, Tech. Rep., 2003. [Online]. Available: http://www4.ncsu.edu/~mtchu/Research/Papers/distance03.pdf
23. P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2008.
24. M. Journée, F. Bach, P.-A. Absil, and R. Sepulchre, "Low-rank optimization on the cone of positive semidefinite matrices," SIAM Journal on Optimization, vol. 20, no. 5, pp. 2327-2351, 2010.
25. I. J. Schoenberg, "Remarks to maurice fréchet's article "sur la d éfinition axiomatique d'une classe d'espace distanciés vectorielle-ment applicable sur l'espace de Hilbert"," Annals of Mathematics, vol. 36, no. 3, pp. 724-732, 1935.
26. B. Vandereycken, P.-A. Absil, and S. Vandewalle, "Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank," in Proceedings of the IEEE 15th Workshop on Statistical Signal Processing, 2009, pp. 389-392.
27. S. Bonnabel and R. Sepulchre, "Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank," SIAM J. Matrix Anal. Appl., vol. 31, pp. 1055-1070, August 2009.
28. G. Meyer, S. Bonnabel, and R. Sepulchre, "Regression on fixed-rank positive semidefinite matrices: a riemannian approach," Journal of Machine Learning Research, vol. 12 (Feb), 2010.
29. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. New York: Springer, 2006.
30. P.-A. Absil, C. G. Baker, and K. A. Gallivan, "Trust-region methods on riemannian manifolds," Foundations of Computational Mathematics, vol. 7, no. 3, pp. 303-330, 2007.
31. S. Burer and R. Monteiro, "A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization," Mathematical Programming, vol. 95, no. 2, pp. 329-357, Feb. 2003.