Folded-concave penalization approaches to tensor completion

Neurocomputing

Volumn 152, Issue , 2015, Pages 261-273

  Cao, Wenfei ^a   Wang, Yao ^a   Yang, Can ^b   Chang, Xiangyu ^a   Han, Zhi ^c   Xu, Zongben ^a  

^a XI'AN JIAOTONG UNIVERSITY   (China)

^b HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

^c SHENYANG INSTITUTE OF AUTOMATION   (China)

Author keywords

Folded concave penalization; Local linear approximation; Nuclear norm; Sparse learning; Tensor completion

Indexed keywords

APPROXIMATION ALGORITHMS; COMPUTATIONAL EFFICIENCY; CONVEX OPTIMIZATION; LAGRANGE MULTIPLIERS; NUMERICAL METHODS;

FOLDED-CONCAVE PENALIZATION; LOCAL LINEAR APPROXIMATION; NUCLEAR NORM; SPARSE LEARNING; TENSOR COMPLETION;

TENSORS;

ARTICLE; CONTROLLED STUDY; FOLDED CONCAVE PENALIZATION; IMAGE PROCESSING; LEARNING ALGORITHM; MACHINE LEARNING; MATHEMATICAL COMPUTING; MATHEMATICAL MODEL; MINMAX CONCAVE PLUS PENALTY MODEL; PROCESS DEVELOPMENT; PROCESS OPTIMIZATION; SMOOTHLY CLIPPED ABSOLUTE DEVIATION PENALTY MODEL; TENSOR COMPLETION;

EID: 84921059478     PISSN: 09252312     EISSN: 18728286     Source Type: Journal    
DOI: 10.1016/j.neucom.2014.10.069     Document Type: Article

References (44)

1. M. Bertalmio, G. Sapiro, V. Caselles, C. Ballester. Image inpainting, in: ACM SIGGRAPH, 2000, pp. 417-424.
2. Korah T., Rasmussen C. Spatio-temporal inpainting for recovering texture maps of occluded building facades. IEEE Trans. Image Process. 2007, 16(7):2262-2271.
3. M. Pauly, N. J. Mitra, J. Giesen, M. Gross, L. Guibas. Example-based 3D scan completion, in: The Symposium on Geometry Processing, 2005, pp. 23-32.
4. Liu J., Musialski P., Wonka P., Ye J. Tensor completion for estimating missing values in visual data. IEEE Trans. Pattern Anal. Mach. Intell. 2013, 35(1):208-220.
5. Kolad T.G., Bader B.W. Tensor decompositions and applications. SIAM Rev. 2009, 51(3):455-500.
6. Candes E.J., Recht B. Exact matrix completion via convex optimization. Found. Comput. Math. 2009, 9(6):717-772.
7. Candes E.J., Tao T. The power of convex relaxation. near-optimal matrix completion. IEEE Trans. Inf. Theory 2009, 56(5):2053-2080.
8. Gandy S., Recht B., Yamada I. Tensor completion and low-n-rank tensor recovery via convex optimization. Inverse Probl. 2011, 27(2):025010.
9. R. Tomioka, H. Kohei, H. Kashima, On the extension of trace norm to tensors, in: NIPS Workshop on Tensors, Kernels, and Machine Learning, 2010.
10. L. Yang, Z. H. Huang, Y. Li. A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery. arXiv preprint, 2013. http://arXiv:1310.1769.
11. Yang L., Zheng Z.H., Shi X.J. A fixed point iterative method for low n-rank tensor pursuit. IEEE Trans. Signal Process. 2013, 61(11):2952-2962.
12. Y. Xu, R. Hao, W. Yin et al. Parallel Matrix factorization for Low-rank tensor completion. arXiv preprint, 2013. http://arXiv:1312.1254.
13. D. Kressner, M. Steinlechner, B. Vandereycken, Low-rank tensor completion by Riemannian optimization, BIT Numer. Math. (2013) 1-22.
14. M. Signoretto, Q.T. Dinh, L.D. Lathauwer, J.A.K. Suykens, Learning with tensors: a framework based on convex optimization and spectral regularization, Mach. Learn. (2013) 1-49.
15. A. Krishnamurthy, A. Singh, Low-rank matrix and tensor completion via adaptive sampling, in: Advances in Neural Information Processing Systems, 2013, pp. 836-844.
16. B. Romera-Paredes, M. Pontil, A new convex relaxation for tensor completion, in: Advances in Neural Information Processing Systems, 2013, pp. 2967-2975.
17. M. Yuan, C. H. Zhang. On Tensor Completion via Nuclear Norm Minimization. arXiv preprint, 2014. http://arXiv:1405.1773.
18. Z. Zhang, G. Ely, S. Aeron, N. Hao, M. Kilmer, Novel factorization strategies for higher order tensors: implications for completion and recovery of multilinear data, arXiv preprint, 2013. http://arXiv:1307.0805.
19. A.H. Phan, A. Cichocki, et al., Tensor completion through multiple Kronecker product decomposition, in: ICASSP, 2013, pp. 3233-3237.
20. H. Rauhut, R. Schneider, Z. Stojanac, Tensor completion in hierarchical tensor representations, arXiv preprint, 2014. http://arXiv:1404.3905.
21. Zou H. The adaptive lasso and its oracle properties. J. Am. Stat. Assoc. 2006, 101(476):1418-1429.
22. Leng C., Lin Y., Wahba G. A note on the lasso and related procedures in model selection. Stat. Sin. 2006, 16:1273-1284.
23. Fan J., Li R. Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 2001, 96(456):1348-1360.
24. Bunea F., She Y., Wegkamp M. Optimal selection of reduced rank estimators of high-dimensional matrices. Ann. Stat. 2011, 39(2):1282-1309.
25. 1/2 regularization. a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learn. Syst. 2012, 23(7):1013-1027.
26. Chartrand R. Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process. Lett. 2007, 14(10):707-710.
27. Zhang C. Nearly unbiased variable selection under minimax concave penalty. Ann. Stat. 2010, 38(2):894-942.
28. Zhang C., Zhang T. A general theory of concave regularization for high dimensional sparse estimation problems. Stat. Sci. 2012, 27(4):576-593.
29. Fan J., Xue L., Zou H. Strong oracle optimality of folded concave penalized estimation. Ann. Stat. 2014, 42(3):819-849.
30. S. Wang, D. Liu, Z. Zhang, Nonconvex relaxation approaches to robust matrix recovery, in: Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence, 2013, pp. 1764-1770.
31. D. Liu, T. Zhou, Q. Qian, C. Xu, Z. Zhang, A nearly unbiased matrix completion approach, in: Proceedings of ECML/PKDD, 2013, pp. 210-225.
32. Cai J.F., Candes E.J., Shen Z. A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 2010, 20(4):1956-1982.
33. Mazumder R., Hastie T., Tibshirani R. Spectral regularization algorithms for learning large incomplete matrices. J. Mach. Learn. Res. 2010, 11(2):2287-2322.
34. Lange K., Hunter D., Yang I. Optimization transfer using surrogate objective functions (with discussion). J. Comput. Graph. Stat. 2000, 9(1):1-20.
35. Hunter R., Li R. Variable selection using MM algorithm. Ann. Stat. 2005, 33(4):1617-1642.
36. Lange K. A gradient algorithm locally equivalent to the EM algorithm. J. R. Stat. Soc.: Ser. B 1995, 57(2):425-437.
37. Bickel P. One-step Huber estimates in the linear model. J. Am. Stat. Assoc. 1975, 70(350):428-434.
38. Z. Lin, M. Chen, Y. Ma, The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices, arXiv preprint, 2010. http://arXiv:1009.5055.
39. Blanchard J.D., Cartis C., Tanner J., Thompson A. Phase transitions for greedy sparse approximation algorithms. Appl. Comput. Harmonic Anal. 2011, 30(2):188-203.
40. Candes E.J., Li X., Ma Y., Wright J. Robust principal component analysis?. J. ACM 2011, 58(3):1-37.
41. Goldfarb D., Qin Z. Robust low rank tensor recovery. models and algorithms. SIAM J. Matrix Anal. Appl. 2014, 35(1):225-253.
42. http://www.media.xiph.org/video/derf/.
43. http://www1.cs.columbia.edu/CAVE/databases/multispectral/.
44. http://www.osirix-viewer.com/datasets/.