Three-Way Tensor Decompositions: A Generalized Minimum Noise Subspace Based Approach

Le Trung Thanh, Viet-Dung Nguyen, Nguyen Linh-Trung, Karim Abed-Meraim

Abstract


Tensor decomposition has recently become a popular method of multi-dimensional data analysis in various applications. The main interest in tensor decomposition is for dimensionality reduction, approximation or subspace purposes. However, the emergence of “big data” now gives rise to increased computational complexity for performing tensor decomposition. In this paper, motivated by the advantages of the generalized minimum noise subspace (GMNS) method, recently proposed for array processing, we proposed two algorithms for principal subspace analysis (PSA) and two algorithms for tensor decomposition using parallel factor analysis (PARAFAC) and higher-order singular value decomposition (HOSVD). The proposed decomposition algorithms can preserve several desired properties of PARAFAC and HOSVD while substantially reducing the computational complexity. Performance comparisons of PSA and tensor decomposition of our proposed algorithms against the state-of-the-art ones were studied via numerical experiments. Experimental results indicated that the proposed algorithms are of practical values.

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References


M. Chen, S. Mao, and Y. Liu, “Big data: A survey,” Mobile networks and applications, vol. 19, no. 2, pp. 171–209, 2014.

E. Acar, C. Aykut-Bingol, H. Bingol, R. Bro, and B. Yener, “Multiway analysis of epilepsy tensors,” Bioinformatics, vol. 23, no. 13, pp. i10–i18, 2007.

C.-F. V. Latchoumane, F.-B. Vialatte, J. Solé-Casals,M. Maurice, S. R. Wimalaratna, N. Hudson, J. Jeong, and A. Cichocki, “Multiway array decomposition analysis of EEGs in Alzheimer’s disease,” Journal of neuroscience methods, vol. 207, no. 1, pp. 41–50, 2012.

F. Cong, Q.-H. Lin, L.-D. Kuang, X.-F. Gong, P. Astikainen, and T. Ristaniemi, “Tensor decomposition of EEG signals: a brief review,” Journal of neuroscience methods, vol. 248, pp. 59–69, 2015.

V. D. Nguyen, K. Abed-Meraim, and N. Linh-Trung, “Fast tensor decompositions for big data processing,” in 2016 International Conference on Advanced Technologies for Communications (ATC), Oct 2016, pp. 215–221.

N. D. Sidiropoulos, L. D. Lathauwer, X. Fu, K. Huang, E. E. Papalexakis, and C. Faloutsos, “Tensor decomposition for signal processing and machine learning,” IEEE Transactions on Signal Processing, vol. 65, no. 13, pp. 3551–3582, July 2017.

T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,” SIAM review, vol. 51, no. 3, pp. 455–500, 2009.

L. Tran, C. Navasca, and J. Luo, “Video detection anomaly via low-rank and sparse decompositions,” in 2012 Western New York Image Processing Workshop (WNYIPW). IEEE, 2012, pp. 17–20.

X. Zhang, X. Shi, W. Hu, X. Li, and S. Maybank, “Visual tracking via dynamic tensor analysis with mean update,” Neurocomputing, vol. 74, no. 17, pp. 3277–3285, 2011.

H. Li, Y. Wei, L. Li, and Y. Y. Tang, “Infrared moving target detection and tracking based on tensor locality preserving projection,” Infrared Physics & Technology, vol. 53, no. 2, pp. 77–83, 2010.

S. Bourennane, C. Fossati, and A. Cailly, “Improvement of classification for hyperspectral images based on tensor modeling,” IEEE Geoscience and Remote Sensing Letters, vol. 7, no. 4, pp. 801–805, 2010.

N. Renard and S. Bourennane, “Dimensionality reduction based on tensor modeling for classification methods,” IEEE Transactions on Geoscience and Remote Sensing, vol. 47, no. 4, pp. 1123–1131, 2009.

H. Fanaee-T and J. Gama, “Event detection from traffic tensors: A hybrid model,” Neurocomputing, vol. 203, pp. 22–33, 2016.

V. D. Nguyen, K. Abed-Meraim, N. Linh-Trung, and R. Weber, “Generalized minimum noise subspace for array processing,” IEEE Transactions on Signal Processing, vol. 65, no. 14, pp. 3789–3802, July 2017.

A. H. Phan and A. Cichocki, “PARAFAC algorithms for large-scale problems,” Neurocomputing, vol. 74, no. 11, pp. 1970–1984, 2011.

A. L. de Almeida and A. Y. Kibangou, “Distributed computation of tensor decompositions in collaborative networks,” in 2013 IEEE 5th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP). IEEE, 2013, pp. 232–235.

A. L. De Almeida and A. Y. Kibangou, “Distributed large-scale tensor decomposition,” in 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2014, pp. 26–30.

V. D. Nguyen, K. Abed-Meraim, and L. T. Nguyen, “Parallelizable PARAFAC decomposition of 3-way tensors,” in 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), April 2015, pp. 5505–5509.

K. Shin, L. Sael, and U. Kang, “Fully scalable methods for distributed tensor factorization,” IEEE Transactions on Knowledge and Data Engineering, vol. 29, no. 1, pp. 100–113, Jan 2017.

D. Chen, Y. Hu, L. Wang, A. Y. Zomaya, and X. Li, “H-PARAFAC: Hierarchical parallel factor analysis of multidimensional big data,” IEEE Transactions on Parallel and Distributed Systems, vol. 28, no. 4, pp. 1091–1104, April 2017.

J. D. Carroll and J.-J. Chang, “Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition,” Psychometrika, vol. 35, no. 3, pp. 283–319, 1970.

N. Halko, P.-G. Martinsson, and J. A. Tropp, “Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions,” SIAM Review, vol. 53, no. 2, pp. 217–288, 2011.

M. W. Mahoney, “Randomized algorithms for matrices and data,” Foundations and Trends R in Machine Learning, vol. 3, no. 2, pp. 123–224, 2011.

D. P. Woodruff, “Sketching as a Tool for Numerical Linear Algebra,” Foundations and Trends in Theoretical Computer Science, vol. 10, no. 1–2, pp. 1–157, 2014.

V. Rokhlin, A. Szlam, and M. Tygert, “A randomized algorithm for principal component analysis,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 3, pp. 1100–1124, 2009.

C. Boutsidis, P. Drineas, and M. Magdon-Ismail, “Near-optimal column-based matrix reconstruction,” SIAM Journal on Computing, vol. 43, no. 2, pp. 687–717, 2014.

A. R. Benson, D. F. Gleich, and J. Demmel, “Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures,” in 2013 IEEE International Conference on Big Data, Oct 2013, pp. 264–272.

N. Kishore Kumar and J. Schneider, “Literature survey on low rank approximation of matrices,” Linear and Multilinear Algebra, vol. 65, no. 11, pp. 2212–2244, 2017.

B. W. Bader, T. G. Kolda et al., “MATLAB Tensor Toolbox Version 2.6,” Available online, February 2015. [Online]. Available: http://www. sandia.gov/ ̃tgkolda/TensorToolbox/

L. De Lathauwer, “A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization,” SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 3, pp. 642–666, 2006.

S. A. Nene, S. K. Nayar, and H. Murase, “Columbia University Image Library (COIL-20),” 1996. [Online]. Available: http://www.cs.columbia.edu/CAVE/ software/ softlib/coil-20.php

M. Haardt, F. Roemer, and G. Del Galdo, “Higher-order SVD-based subspace estimation to improve the parameter estimation accuracy in multidimensional harmonic retrieval problems,” IEEE Transactions on Signal Processing, vol. 56, no. 7, pp. 3198–3213, 2008.

N. Vannieuwenhoven, R. Vandebril, and K. Meerbergen, “A new truncation strategy for the higher-order singular value decomposition,” SIAM Journal on Scientific Computing, vol. 34, no. 2, pp. A1027–A1052, 2012.




DOI: http://dx.doi.org/10.21553/rev-jec.196

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