Wedderburn rank reduction and Krylov subspace method for tensor approximation. Part 1: Tucker case

SIAM Journal on Scientific Computing

Volumn 34, Issue 1, 2012, Pages

  Goreinov, S A ^a   Oseledets, I V ^a   Savostyanov, D V ^a  

^a INSTITUTE OF NUMERICAL MATHEMATICS   (Russian Federation)

Author keywords

Fast compression; Krylov subspace methods; Multidimensional arrays; Sparse tensors; Structured tensors; Tucker approximation; Wedderburn rank reduction

Indexed keywords

APPROXIMATION ALGORITHMS; MATRIX ALGEBRA; NUMERICAL METHODS; QUANTUM CHEMISTRY; VECTORS;

FAST COMPRESSIONS; KRYLOV-SUBSPACE METHODS; MATRIX CASE; MULTIDIMENSIONAL ARRAYS; SPARSE TENSORS; STRUCTURED TENSORS; TENSOR APPROXIMATION; TUCKER APPROXIMATIONS; VECTOR MULTIPLICATION; WEDDERBURN RANK REDUCTIONS;

TENSORS;

EID: 84861364912     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/100792056     Document Type: Article

References (41)

1. B. W. Bader and T. G. Kolda, Efficient MATLAB computations with sparse and factored tensors, SIAM J. Sci. Comput., 30 (2007), pp. 205-231.
2. A. Bouaricha, Tensor-Krylov methods for large nonlinear equations, Comput. Optim. Appl., 5 (1996), pp. 207-232. (Pubitemid 126708360)
3. J. D. Caroll and J. J Chang, Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition, Psychometrika, 35 (1970), pp. 283-319.
4. S. R. Chinnamsetty, H.-J. Flad, V. Khoromskaia, and B. N. Khoromskij, Tensor decomposition in electronic structure calculations on 3D Cartesian grids, J. Comput. Phys., 228 (2009), pp. 5749-5762.
5. M. T. Chu, R. E. Funderlic, and G. H. Golub, A rank-one reduction formula and its applications to matrix factorizations, SIAM Rev., 37 (1995), pp. 512-530.
6. L. De Lathauwer, B. De Moor, and J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253-1278. (Pubitemid 33217068)
7. L. De Lathauwer, B. De Moor, and J. Vandewalle, On best rank-1 and rank- (R1,R2, . . . ,RN) approximation of high-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1324-1342. (Pubitemid 33217072)
8. 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.
9. A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland, Amsterdam, 1993.
10. L. Eldén 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.
11. H.-J. Flad, B. N. Khoromskij, D. V. Savostyanov, and E. E. Tyrtyshnikov, Verification of the cross 3D algorithm on quantum chemistry data, Russian J. Numer. Anal. Math. Modelling, 23 (2008), pp. 329-344.
12. G. Golub and W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Numer. Anal., 2 (1965), pp. 205-224.
13. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
14. S. A. Goreinov, I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov, and N. L. Zamarashkin, How to find a good submatrix, in Matrix Methods: Theory, Algorithms, Applications, V. Olshevsky and E. Tyrtyshnikov, eds., World Scientific, Hackensack, NJ, 2010, pp. 247-256.
15. S. A. Goreinov and E. E. Tyrtyshnikov, The maximal-volume concept in approximation by low-rank matrices, Contemp. Math., 208 (2001), pp. 47-51.
16. A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, 1997.
17. R. A. Harshman, Foundations of the Parafac procedure: Models and conditions for an explanatory multimodal factor analysis, UCLA Working Papers in Phonetics, 16 (1970), pp. 1-84.
18. J. Hastad, Tensor rank is NP-complete, J. Algorithms, 11 (1990), pp. 644-654.
19. F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of products, J. Math. Phys, 6 (1927), pp. 164-189.
20. 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.
21. B. N. Khoromskij and V. Khoromskaia, Multigrid accelerated tensor approximation of function related multidimensional arrays, SIAM J. Sci. Comput., 31 (2009), pp. 3002-3026.
22. B. N. Khoromskij, V. Khoromskaia, and H.-J. Flad, Numerical solution of the Hartree- Fock equation in multilevel tensor-structured format, SIAM J. Sci. Comput., 33 (2011), pp. 45-65.
23. T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500.
24. P. M. Kroonenberg and J. de Leeuw, Principal component analysis of three-mode data by means of alternating least squares algorithms, Psychometrika, 45 (1980), pp. 69-97.
25. I. V. Oseledets, Approximation of 2d × 2d matrices using tensor decomposition, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2130-2145.
26. I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput., 33 (2011), pp. 2295-2317.
27. 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., 30 (2008), pp. 939-956.
28. I. V. Oseledets, D. V. Savostyanov, and E. E. Tyrtyshnikov, Linear algebra for tensor problems, Computing, 85 (2009), pp. 169-188.
29. I. V. Oseledets, D. V. Savostyanov, and E. E. Tyrtyshnikov, Cross approximation in tensor electron density computations, Numer. Linear Algebra Appl., 17 (2010), pp. 935- 952.
30. I. V. Oseledets and E. E. Tyrtyshnikov, Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM J. Sci. Comput., 31 (2009), pp. 3744-3759.
31. C. C. Paige, Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix, IMA J. Appl. Math., 18 (1976), p. 341.
32. B. Savas and L. Eldén, Krylov Subspace Methods for Tensor Computations, Preprint LITHMAT- R-2009-02-SE, Dep. Math., Linköpings Univ., Linköping, 2009.
33. B. Savas and L. Eldén, Krylov-Type Methods for Tensor Computations, Preprint arXiv:1005.0683[math.NA], 2011.
34. D. V. Savostyanov, Fast revealing of mode ranks of tensor in canonical format, Numer. Math. Theor. Methods Appl., 2 (2009), pp. 439-444.
35. D. V. Savostyanov and E. E. Tyrtyshnikov, Approximate multiplication of tensor matrices based on the individual filtering of factors, J. Comput. Math. Math. Phys., 49 (2009), pp. 1662-1677.
36. D. V. Savostyanov, E. E. Tyrtyshnikov, and N. L. Zamarashkin, Fast truncation of mode ranks for bilinear tensor operations, Numer. Linear Algebra Appl., (2011).
37. A. L. Traud, E. D. Kelsic, P. J. Mucha, and M. A. Porter, Comparing community structure to characteristics in online collegiate social networks, SIAM Rev., 53 (2011), pp. 526- 543.
38. L. R. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, 31 (1966), pp. 279-311.
39. E. E. Tyrtyshnikov, Incomplete cross approximation in the mosaic-skeleton method, Computing, 64 (2000), pp. 367-380.
40. J. H. M Wedderburn, Lectures on matrices, in Colloquium Publications 17, AMS, New York, 1934.
41. T. Zhang and G. H. Golub, Rank-one approximation to high order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 534-550.