Riemannian optimization on tensor products of grassmann manifolds: Applications to generalized Rayleigh-quotients

SIAM Journal on Matrix Analysis and Applications

Volumn 33, Issue 1, 2012, Pages 210-234

  Curtef, O ^a   Dirr, G ^a   Helmke, U ^a  

^a UNIVERSITY OF WÜRZBURG   (Germany)

Author keywords

Best approximation of tensors; Conjugated gradient method; Entanglement measure; Grassmann manifold; Multilinear rank; Newton method; Riemannian optimization; Subspace clustering

Indexed keywords

BEST APPROXIMATIONS; CONJUGATED GRADIENT METHOD; ENTANGLEMENT MEASURE; GRASSMANN MANIFOLD; MULTILINEAR RANK; SUBSPACE CLUSTERING;

CLUSTERING ALGORITHMS; DATA COMPRESSION; GRADIENT METHODS; IMAGE PROCESSING; NEWTON-RAPHSON METHOD; OPTIMIZATION; QUANTUM COMPUTERS;

TENSORS;

EID: 84861634339     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/100792032     Document Type: Article

References (30)

1. P. A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2007.
2. R. Bhatia, Partial traces and entropy inequalities, Linear Algebra Appl., 370 (2003), pp. 125-132.
3. R. W. Brockett, Least squares matching problems, Linear Algebra Appl., 122-124 (1989), pp. 761-777.
4. P. Comon, Tensor Decompositions: State of the Art and Applications, Math. Signal Process. 5, Oxford University Press, Oxford, UK, 2002, pp. 1-24.
5. O. Curtef, G. Dirr, and U. Helmke, Riemannian optimization on tensor products of Grassmann manifolds: Applications to generalized Rayleigh-quotients manifolds, arXiv: 1005.4854, 2011.
6. G. Dirr, U. Helmke, M. Kleinsteuber, and T. Schulte-Herbrüggen, Relative C-numerical ranges for applications in quantum control and quantum information, Linear Multilinear Algebra, 56 (2008), pp. 27-51.
7. C. Eckart and G. Young, The approximation of one matrix by another of lower rank, Psychometrika, 1 (1936), pp. 211-263.
8. L. Elden 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.
9. D. Gabay, Minimizing a differentiable function over a differentiable manifold, J. Optim. Theory Appl., 37 (1982), pp. 177-21.
10. K. A. Gallivan, A. Srivastava, X. Liu, and P. Van Dooren, Efficient algorithms for inferences on Grassmann manifolds, in Proceedings of the IEEE Conference on Statistical Signal Processing, 2003, pp. 315-318.
11. U. Helmke, K. Hüper, and J. Trumf, Newton's method on Grassmann manifolds, arXiv: 0709.2205v2, 2007.
12. U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer-Verlag, London, 1994.
13. M. W. Hirsch, Differential Topology, Springer-Verlag, New York, 1976.
14. J. Ho, M. H. Yang, J. Lim, K. C. Lee, and D. Kriegman, Clustering appearances of objects under varying illumination conditions, in Proceedings of the IEEE Computer Vision and Pattern Recognition Conference, 2003, pp. 11-18.
15. M. Ishteva, L. De Lathauwer, P. A. Absil, and S. Van Huffel, Best low multilinear rank approximation of higher-order tensors, based on the Riemannian trust-region scheme, SIAM. J. Matrix Anal. Appl., 32 (2011), pp. 115-135.
16. M. Ishteva, L. De Lathauwer, P. A. Absil, and S. Van Huffel, Differential-geometric Newton method for the best rank-(R1,R2,R3) approximation of tensors, Numer. Algorithms, 51 (2009), pp. 179-194.
17. L. De Lathauwer, B. De Moor, and J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253-1278.
18. 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.
19. Nielsen and Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000.
20. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed., Springer-Verlag, New York, 2006.
21. B. Savas, Algorithm Package Manual: Best Low Rank Tensor Approximation, Department of Mathematics, Linköping University, Linköping, Sweden, 2008; also available online from http://www.mai.liu.se/-besav/soft.html.
22. B. Savas and L.-H. Lim, Quasi-Newton methods on Grassmannians and multilinear approximations of tensors, SIAM J. Sci. Comput., 32 (2010), pp. 3352-3393.
23. M. Shub, Some remarks on dynamical systems and numerical analysis, Dynam. Systems Partial Differential Equations (Caracas), (1986), pp. 69-91.
24. 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.
25. T. Smith, Optimization Techniques on Riemannian Manifolds, Fields Inst. Commun., 3, American Mathematical Society, Providence, RI, 1994.
26. M. Tipping and C. Bishop, Mixtures of probabilistic principal component analyzers, Neural Computation, 11 (1999).
27. L. R. Tucker, Some mathematical notes of three-mode factor analysis, Psychometrika, 31 (1966), pp. 279-311.
28. C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds, Kluwer Academic, Dordrecht, The Netherlands, 1994.
29. R. Vidal, Y. Ma, and J. Piazzi, A new GPCA algorithm for clustering subspaces fitting, differentiating and dividing polynomials, in Proceedings of the IEEE Computer Vision and Pattern Recognition Conference, 2004.
30. T. Wei and P. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A, 68 (2003), 042307.