Breaking the curse of dimensionality using decompositions of incomplete tensors: Tensor-based scientific computing in big data analysis

IEEE Signal Processing Magazine

Volumn 31, Issue 5, 2014, Pages 71-79

  Vervliet, Nico ^a   Debals, Otto ^a   Sorber, Laurent ^a   De Lathauwer, Lieven ^b  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b Department of Engineering and Technology ^*  (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

ARRAY PROCESSING; BIG DATA; COMPUTATION THEORY; HIGHER ORDER STATISTICS; INFORMATION THEORY; MATRIX ALGEBRA; QUANTUM OPTICS; QUANTUM THEORY; SIGNAL PROCESSING;

COMPRESSIVE SENSING; CURSE OF DIMENSIONALITY; DECOMPOSITION ALGORITHM; HIGHER-ORDER TENSOR; MEMORY REQUIREMENTS; MULTIVARIATE FUNCTION; QUANTUM INFORMATION THEORY; TENSOR DECOMPOSITION;

TENSORS;

EID: 85032751253     PISSN: 10535888     EISSN: None     Source Type: Journal    
DOI: 10.1109/MSP.2014.2329429     Document Type: Article

References (37)

1. P. Comon and C. Jutten, Handbook of Blind Source Separation: Independent Component Analysis and Applications. New York: Academic, 2010.
2. N. Sidiropoulos, R. Bro, and G. Giannakis, "Parallel factor analysis in sensor array processing," IEEE Trans. Signal Processing, vol. 48, no. 8, pp. 2377-2388, 2000.
3. I. Oseledets, "Tensor-train decomposition," SIAM J. Sci. Comput., vol. 33, no. 5, pp. 2295-2317, 2011.
4. B. Khoromskij, "Tensors-structured numerical methods in scientific computing: Survey on recent advances," Chemomet. Intell. Lab. Syst., vol. 110, no. 1, pp. 1-19, 2012.
5. L. Grasedyck, D. Kressner, and C. Tobler. (2013, Feb.). A literature survey of low-rank tensor approximation techniques. [Online]. Available: http://arxiv.org/pdf/1302.7121.pdf
6. W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus (Springer Series in Computational Mathematics, vol. 42). Heidelberg: Springer, 2012.
7. R. Orus. (2014, June). A practical introduction to tensor networks: Matrix product states and projected entangled pair states. [Online]. Available: http://arxiv.org/pdf/1306.2164.pdf
8. I. Oseledets and E. Tyrtyshnikov, "TT-cross approximation for multidimensional arrays," Linear Algebra Appl., vol. 432, no. 1, pp. 70-88, 2010.
9. T. Kolda and B. Bader, "Tensor decompositions and applications," SIAM Rev., vol. 51, no. 3, pp. 455-500, 2009.
10. A. Cichocki, D. Mandic, C. Caiafa, A.-H. Phan, G. Zhou, Q. Zhao, and L. De Lathauwer, "Tensor decompositions for signal processing applications," IEEE Signal Processing Mag., to be published.
11. P. Comon, "Tensors: A brief introduction," IEEE Signal Processing Mag., vol. 31, no. 3, pp. 44-53, May 2014.
12. I. Domanov and L. De Lathauwer, "On the uniqueness of the canonical polyadic decomposition of third-order tensors-Part II: Uniqueness of the overall decomposition," SIAM J. Matrix Anal. Appl., vol. 34, no. 3, pp. 876-903, 2013.
13. G. Beylkin and M. J. Mohlenkamp, "Numerical operator calculus in higher dimensions," Proc. Natl. Acad. Sci., vol. 99, no. 16, pp. 10246-10251, 2002.
14. W. Krijnen, T. Dijkstra, and A. Stegeman, " On the non-existence of optimal solutions and the occurrence of "degeneracy" in the CANDECOMP/PARAFAC model," Psychometrika, vol. 73, no. 3, pp. 431-439, 2008.
15. A. Smilde, R. Bro, P. Geladi, and J. Wiley, Multi-Way Analysis with Applications in the Chemical Sciences. Chichester, U.K.: Wiley, 2004.
16. L. Sorber, M. Van Barel, and L. De Lathauwer, "Structured data fusion," Tech. Rep. 13-177, ESAT-STADIUS, KU Leuven, Belgium, 2013.
17. L. De Lathauwer, B. De Moor, and J. Vandewalle, "A multilinear singular value decomposition," SIAM J. Matrix Anal. Appl., vol. 21, no. 4, pp. 1253-1278, 2000. (Pubitemid 33217068)
18. I. Oseledets and E. Tyrtyshnikov, "Breaking the curse of dimensionality, or how to use SVD in many dimensions," SIAM J. Sci. Comput., vol. 31, no. 5, pp. 3744-3759, 2009.
19. E. Acar, D. Dunlavy, T. Kolda, and M. Mørup, "Scalable tensor factorizations for incomplete data," Chemomet. Intell. Lab. Syst., vol. 106, no. 1, pp. 41-56, 2011.
20. S. Gandy, B. Recht, and I. Yamada, "Tensor completion and low-n-rank tensor recovery via convex optimization," Inverse Prob., vol. 27, no. 2, p. 025010, 2011.
21. E. Candès and M. Wakin, "An introduction to compressive sampling," IEEE Signal Processing Mag., vol. 25, no. 2, pp. 21-30, 2008.
22. C. F. Caiafa and A. Cichocki, "Multidimensional compressed sensing and their applications," Wiley Interdisciplinary Rev.: Data Mining Knowl. Discov., vol. 3, no. 6, pp. 355-380, 2013.
23. N. Sidiropoulos and A. Kyrillidis, "Multi-way compressed sensing for sparse lowrank tensors," IEEE Signal Process. Lett., vol. 19, no. 11, pp. 757-760, Nov. 2012.
24. Q. Li, D. Schonfeld, and S. Friedland, "Generalized tensor compressive sensing," in Proc. 2013 IEEE Int. Conf. Multimedia and Expo (ICME), July 2013, pp. 1-6.
25. L. Sorber, M. Van Barel, and L. De Lathauwer, "Optimization-based algorithms for tensor decompositions: Canonical polyadic decomposition, decomposition in rank-(Lr, Lr, 1) terms, and a new generalization," SIAM J. Optim., vol. 23, no. 2, pp. 695-720, 2013.
26. E. Acar, D. Dunlavy, and T. Kolda, "A scalable optimization approach for fitting canonical tensor decompositions," J. Chemomet., vol. 25, no. 2, pp. 67-86, 2011.
27. A.-H. Phan, P. Tichavsky, and A. Cichocki, "Low complexity damped Gauss-Newton algorithms for CANDECOMP/PARAFAC," SIAM J. Appl. Math., vol. 34, no. 1, pp. 126-147, 2013.
28. O. Debals and N. Vervliet, "Efficiënte tensorgebaseerde methoden voor modellering en signaalscheiding," (in Dutch), Master's thesis, Departments of Comput. Sci. and Electr. Eng., KU Leuven, 2013.
29. S. Goreinov, N. Zamarashkin, and E. Tyrtyshnikov, "Pseudo-skeleton approximations by matrices of maximal volume," Math. Notes, vol. 62, no. 4, pp. 515-519, 1997. (Pubitemid 127725193)
30. E. Tyrtyshnikov, "Incomplete cross approximation in the mosaic-skeleton method," Computing, vol. 64, no. 4, pp. 367-380, 2000.
31. I. Oseledets, D. Savostianov, and 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.
32. C. Caiafa and A. Cichocki, "Generalizing the column-row matrix decomposition to multi-way arrays," Linear Algebra Appl., vol. 433, no. 3, pp. 557-573, 2010.
33. M. Mahoney, M. Maggioni, and P. Drineas, "Tensor-CUR decompositions for tensor-based data," SIAM J. Matrix Anal. Appl., vol. 30, no. 3, pp. 957-987, 2008.
34. X. Liu, N. D. Sidiropoulos, and T. Jiang, "Multidimensional harmonic retrieval with applications in MIMO wireless channel sounding," in Space-Time Processing for MIMO Communications, A. Gershman and N. Sidiropoulos, Eds. Hoboken, NJ: Wiley, 2005.
35. M. Sorensen and L. De Lathauwer, "Multidimensional harmonic retrieval via coupled canonical polyadic decomposition," Internal Rep. 13-240, ESAT-STADIUS, KU Leuven, Belgium, 2013.
36. S. Van Huffel, H. Chen, C. Decanniere, and P. Vanhecke, "Algorithm for timedomain NMR data fitting based on total least squares," J. Magn. Reson., Ser. A, vol. 110, no. 2, pp. 228-237, 1994.
37. L. Sorber, M. Van Barel, and L. De Lathauwer. (2014, Jan.). Tensorlab v2.0. [Online]. Available: www.tensorlab.net