May  2015, 9(2): 601-624. doi: 10.3934/ipi.2015.9.601

Parallel matrix factorization for low-rank tensor completion

1. 

Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005

2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian, China, China

3. 

Department of Mathematics, University of California, Los Angeles, CA 90095-1555

Received  December 2013 Revised  October 2014 Published  March 2015

Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data reconstruction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode matricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known.
    Phase transition plots reveal that our algorithm can recover a variety of synthetic low-rank tensors from significantly fewer samples than the compared methods, which include a matrix completion method applied to tensor recovery and two state-of-the-art tensor completion methods. Further tests on real-world data show similar advantages. Although our model is non-convex, our algorithm performs consistently throughout the tests and gives better results than the compared methods, some of which are based on convex models. In addition, subsequence convergence of our algorithm can be established in the sense that any limit point of the iterates satisfies the KKT condtions.
Citation: Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601
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show all references

References:
[1]

SIAM J. Optim., 20 (2010), 1956-1982. doi: 10.1137/080738970.  Google Scholar

[2]

Foundations of Computational Mathematics, 9 (2009), 717-772. doi: 10.1007/s10208-009-9045-5.  Google Scholar

[3]

IMA Journal of Numerical Analysis, 32 (2012), 227-245. doi: 10.1093/imanum/drq039.  Google Scholar

[4]

Inverse Problems, 27 (2011), 025010, 1-19. doi: 10.1088/0266-5611/27/2/025010.  Google Scholar

[5]

Oper. Res. Lett., 26 (2000), 127-136. doi: 10.1016/S0167-6377(99)00074-7.  Google Scholar

[6]

Mathematical Programming, (2014), 1-35. doi: 10.1007/s10107-014-0774-0.  Google Scholar

[7]

Journal of Chemometrics, 14 (2000), 105-122. doi: 10.1002/1099-128X(200005/06)14:3<105::AID-CEM582>3.0.CO;2-I.  Google Scholar

[8]

SIAM Journal on Matrix Analysis and Applications, 34 (2013), 148-172. doi: 10.1137/110837711.  Google Scholar

[9]

SIAM review, 51 (2009), 455-500. doi: 10.1137/07070111X.  Google Scholar

[10]

in Data Mining, Fifth IEEE International Conference on, IEEE, 2005. doi: 10.1109/ICDM.2005.77.  Google Scholar

[11]

Geophysics, 77 (2012), V113-V122. doi: 10.1190/geo2011-0399.1.  Google Scholar

[12]

BIT, 54 (2014), 447-468. doi: 10.1007/s10543-013-0455-z.  Google Scholar

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[14]

in 2010 17th IEEE International Conference on Image Processing (ICIP), IEEE, 2010, 517-520. doi: 10.1109/ICIP.2010.5651225.  Google Scholar

[15]

in 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012, 2925-2928. doi: 10.1109/ICASSP.2012.6288528.  Google Scholar

[16]

IEEE Transactions on Pattern Analysis and Machine Intelligence, (2012), 208-220. doi: 10.1109/TPAMI.2012.39.  Google Scholar

[17]

Mathematical Programming, 128 (2011), 321-353. doi: 10.1007/s10107-009-0306-5.  Google Scholar

[18]

preprint, arXiv:1307.5870, (2013). Google Scholar

[19]

IEEE Transactions on Image Processing, 16 (2007), 545-553. doi: 10.1109/TIP.2006.888343.  Google Scholar

[20]

SIAM Review, 52 (2010), 471-501. doi: 10.1137/070697835.  Google Scholar

[21]

preprint, arXiv:1307.4653, (2013). Google Scholar

[22]

Nuclear Science, IEEE Transactions on, 46 (1999), 2075-2084. doi: 10.1109/23.819285.  Google Scholar

[23]

in Proceedings of the 14th international conference on World Wide Web, ACM, 2005, 382-390. doi: 10.1145/1060745.1060803.  Google Scholar

[24]

Pacific Journal of Optimization, 6 (2010), 615-640.  Google Scholar

[25]

Journal of Optimization Theory and Applications, 109 (2001), 475-494. doi: 10.1023/A:1017501703105.  Google Scholar

[26]

Psychometrika, 31 (1966), 279-311. doi: 10.1007/BF02289464.  Google Scholar

[27]

Mathematical Programming Computation, 4 (2012), 333-361. doi: 10.1007/s12532-012-0044-1.  Google Scholar

[28]

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[29]

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[30]

arXiv:1410.1386, (2014). Google Scholar

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preprint, arXiv:1307.0805v3, (2013). Google Scholar

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