Fast adaptive interpolation of multi-dimensional arrays in tensor train format

2011 7th International Workshop on Multidimensional (nD) Systems, nDS 2011

Volumn , Issue , 2011, Pages

  Savostyanov, Dmitry ^a   Oseledets, Ivan ^a  

^a INSTITUTE OF NUMERICAL MATHEMATICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

ADAPTIVE INTERPOLATION; APPROXIMATION SCHEME; DENSE ARRAYS; FAST INTERPOLATION; MULTIDIMENSIONAL ARRAYS; NUMERICAL EXAMPLE; UNIFORM GRIDS;

INTERPOLATION; QUANTUM CHEMISTRY;

TENSORS;

EID: 83055196753     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/nDS.2011.6076873     Document Type: Conference Paper

References (30)

1. F. L. Hitchcock, "The expression of a tensor or a polyadic as a sum of products," J. Math. Phys, vol. 6, no. 1, pp. 164-189, 1927.
2. J. D. Caroll and J. J. Chang, "Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition," Psychometrika, vol. 35, pp. 283-319, 1970.
3. R. A. Harshman, "Foundations of the Parafac procedure: models and conditions for an explanatory multimodal factor analysis," UCLA Working Papers in Phonetics, vol. 16, pp. 1-84, 1970.
4. T. G. Kolda and B. W. Bader, "Tensor decompositions and applications,"SIAM Review, vol. 51, no. 3, pp. 455-500, 2009.
5. J. H̊astad, "Tensor rank is NP-complete," Journal of Algorithms, vol. 11, no. 4, pp. 644-654, 1990.
6. V. de Silva and L. Lim, "Tensor rank and the ill-posedness of the best low-rank approximation problem," SIAM J. Matrix Anal. Appl., vol. 30, no. 3, pp. 1084-1127, 2008.
7. R. Bro, "PARAFAC: Tutorial and applications," Chemometrics and Intelligent Lab. Syst., vol. 38, no. 2, pp. 149-171, 1997.
8. L. de Lathauwer, B. de Moor, and J. Vandewalle, "Computing of canonical decomposition by means of a simultaneous generalized Schur decomposition," SIAM J. Matrix Anal. Appl., vol. 26, pp. 295-327, 2004. (Pubitemid 40541662)
9. I. V. Oseledets, D. V. Savostyanov, and E. E. Tyrtyshnikov, "Fast simultaneous orthogonal reduction to triangular matrices," SIAM J. Matrix Anal. Appl., vol. 31, no. 2, pp. 316-330, 2009.
10. L. R. Tucker, "Some mathematical notes on three-mode factor analysis,"Psychometrika, vol. 31, pp. 279-311, 1966.
11. L. de Lathauwer, B. de Moor, and J. Vandewalle, "A multilinear singular value decomposition," SIAM J. Matrix Anal. Appl., vol. 21, pp. 1253-1278, 2000.
12. N) approximation of high-order tensors," SIAM J. Matrix Anal. Appl., vol. 21, pp. 1324-1342, 2000.
13. I. V. Oseledets, "Tensor train decomposition," SIAM J. Sci. Comp, 2011.
14. -, "Compact matrix form of the d-dimensional tensor decomposition," INM RAS, Moscow, Preprint 2009-01, 2009. [Online]. Available: http://pub.inm.ras.ru
15. I. V. Oseledets and E. E. Tyrtyshnikov, "Breaking the curse of dimensionality, or how to use svd in many dimensions," SIAM J. Sci. Comp., vol. 31, no. 5, pp. 3744-3759, 2009.
16. -, "TT-cross algorithm for the approximation of multidimensional arrays," Linear Algebra Appl., vol. 432, no. 1, pp. 70-88, 2010.
17. E. E. Tyrtyshnikov, "Incomplete cross approximation in the mosaic-skeleton method," Computing, vol. 64, no. 4, pp. 367-380, 2000.
18. M. Bebendorf, "Approximation of boundary element matrices," Numerische Mathematik, vol. 86, no. 4, pp. 565-589, 2000.
19. S. A. Goreinov and E. E. Tyrtyshnikov, "The maximal-volume concept in approximation by low-rank matrices," Contemporary Mathematics, vol. 208, pp. 47-51, 2001.
20. S. Goreinov, I. Oseledets, D. Savostyanov, E. Tyrtyshnikov, and N. Zamarashkin, "How to find a good submatrix," in Matrix Methods: Theory, Algorithms, Applications, V. Olshevsky and E. Tyrtyshnikov, Eds. World Scientific Publishing, 2010, pp. 247-256.
21. I. V. Oseledets, D. V. Savostianov, and E. E. Tyrtyshnikov, "Tucker dimensionality reduction of three-dimensional arrays in linear time,"SIAM J. Matrix Anal. Appl., vol. 30, no. 3, pp. 939-956, 2008.
22. S. White, "Density-matrix algorithms for quantum renormalization groups," Physical Review B, vol. 48, no. 14, pp. 10 345-10 356, 1993.
23. d matrices using tensor decomposition," SIAM J. Matrix Anal. Appl., vol. 31, no. 4, pp. 2130-2145, 2010.
24. B. N. Khoromskij, "O(d log n)-Quantics approximation of N-d tensors in high-dimensional numerical modeling," Constr. Appr., 2011.
25. M. Espig, L. Grasedyck, and W. Hackbusch, "Black box low tensor rank approximation using fibre-crosses," Constr. Appr., vol. 30, no. 3, pp. 557-597, 2009.
26. W. Hackbusch and S. Kühn, "A new scheme for the tensor representation,"J. Fourier Anal. Appl., vol. 15, no. 5, pp. 706-722, 2009.
27. L. Grasedyck, "Hierarchical singular value decomposition of tensors,"SIAM J. Matrix Anal. Appl., vol. 31, no. 4, pp. 2029-2054, 2010.
28. J. Ballani, L. Grasedyck, and M. Kluge, "Black box approximation of tensors in Hierarchical Tucker format,"MPI MIS, Leipzig, Preprint 57, 2010. [Online]. Available: www.mis.mpg.de/preprints/2010/preprint2010 57.pdf
29. G. Vidal, "Efficient classical simulation of slightly entangled quantum computations," Physical Review Letters, vol. 91, no. 14, p. 147902, 2003.
30. S. Ostlund and S. Rommer, "Thermodynamic limit of density matrix renormalization," Physical review letters, vol. 75, no. 19, pp. 3537-3540, 1995.