Recovery of Corrupted Low-Rank Tensors

Full-Text HTML XML Download Download as PDF (Size:647KB) PP. 229-244
DOI: 10.4236/am.2017.82019    297 Downloads   378 Views  


This paper studies the problem of recovering low-rank tensors, and the tensors are corrupted by both impulse and Gaussian noise. The problem is well accomplished by integrating the tensor nuclear norm and the l1-norm in a unified convex relaxation framework. The nuclear norm is adopted to explore the low-rank components and the l1-norm is used to exploit the impulse noise. Then, this optimization problem is solved by some augmented-Lagrangian-based algorithms. Some preliminary numerical experiments verify that the proposed method can well recover the corrupted low-rank tensors.

Cite this paper

Fan, H. and Kuang, G. (2017) Recovery of Corrupted Low-Rank Tensors. Applied Mathematics, 8, 229-244. doi: 10.4236/am.2017.82019.


[1] Jolliffe, I. (2002) Principal Component Analysis. Wiley Online Library.
[2] Candes, E.J., Li, X.D., Ma, Y. and Wright, J. (2011) Robust Principal Component Analysis? Journal of the ACM, 58, Article Number: 11.
[3] Chandrasekaran, V., Sanghavi, S., Parrilo, P.A. and Willsky, A.S. (2011) Rank-Sparsity Incoherence for Matrix Decomposition. SIAM Journal on Optimization, 21, 572-596.
[4] Wright, J., Ganesh, A., Rao, S.Y.P. and Ma, Y. (2009) Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization Neural Information Processing Systems (NIPS).
[5] Tao, M. and Yuan, X. (2011) Recovering Low-Rank and Sparse Components of Matrices from Incomplete and Noisy Observations. SIAM Journal on Optimization, 21, 57-81.
[6] Candes, E.J. and Recht, B. (2009) Exact Matrix Completion via Convex Optimization. Foundations of Computational Mathematics, 9, 717-772.
[7] Candes, E.J. and Tao, T. (2010) The Power of Convex Relaxation: Near-Optimal Matrix Completion. IEEE Transaction on Information Theory, 56, 2053-2080.
[8] Wang, H. and Ahuja, N. (2004) Compact Representation of Multidimensional Data Using Tensor Rank-One Decomposition.
[9] Bloy, L. and Verma, R. (2008) On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function. 11th International Conference on Medical Image Computing and Computer-Assisted Intervention, New York, 6-10 September 2008, 1-8.
[10] Ghosh, A., Tsigaridas, E., Descoteaux, M., Comon, P., Mourrain, B. and Deriche, R. (2008) A Polynomial Based Approach to Extract the Maxima of an Antipodally Symmetric Spherical Function and Its Application to Extract Fiber Directions from the Orientation Distribution Function in Diffusion MRI.
[11] Qi, L., Yu, G. and Wu, E.X. (2010) Higher Order Positive Semi-Definite Diffusion Tensor Imaging. SIAM Journal on Imaging Sciences, 3, 416-433.
[12] Hilling, J.J. and Sudbery, A. (2010) The Geometric Measure of Multipartite Entanglement and the Singular Values of a Hypermatrix. Journal of Mathematical Physics, 51, Article ID: 072102.
[13] Hu, S. and Qi, L. (2012) Algebraic Connectivity of an Even Uniform Hypergraph. Journal of Combinatorial Optimization, 24, 564-579.
[14] Li, W. and Ng, M. (2011) Existence and Uniqueness of Stationary Probability Vector of a Transition Probability Tensor Technical Report. Department of Mathematics, the Hong Kong Baptist University, Hong Kong.
[15] Liu, J., Musialski, P., Wonka, P. and Ye, J. (2009) Tensor Completion for Estimating Missing Values in Visual Data In ICCV.
[16] Lubich, C. (2008) From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis EMS Zürich.
[17] Beck, M.H., Jackle, A., Worth, G.A. and Meyer, H.D. (1999) The Multiconfiguration Time-Dependent Hartree (MCTDH) Method: A Highly Efficient Algorithm for Propagating Wavepackets. Physics Reports, 324, 1-105.
[18] Wang, H. and Thoss, M. (2009) Numerically Exact Quantum Dynamics for Indistinguishable Particles: The Multilayer Multiconfiguration Time-Dependent Hartree Theory in Second Quantization Representation. Journal of Chemical Physics, 131, Article ID: 024114.
[19] Paredes, B.R., Aung, H., Berthouze, N.B. and Pontil, M. (2013) Multilinear Multitask Learning. Journal of Machine Learning Research, 28, 1444-1452.
[20] Kolda, T.G. and Bader, B.W. (2009) Tensor Decompositions and Applications. SIAM Review, 51, 455-500.
[21] Zhang, Z., Ely, G., Aeron, S., Hao, N. and Kilmer, M.E. (2014) Novel Methods for Multilinear Data Completion and De-Noising Based on Tensor-SVD. Computer Science, 44, 3842-3849.
[22] Semerci, O., Hao, N., Kilmer, M.E. and Miller, E.L. (2013) Tensor-Based Formulation and Nuclear Norm Regularization for Multienergy Computed Tomography. IEEE Transactions on Image Processing, 23, 1678-1693.
[23] Lu, C., Feng, J., Chen, Y., Liu, W., Lin, Z. and Yan, S. (2016) Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization.
[24] Cai, X., Han, D. and Yuan, X. (2014) The Direct Extension of ADMM for Three-Block Separable Convex Minimization Models Is Convergent When One Function Is Strongly Convex.
[25] Li, M., Sun, D. and Toh, K. (2014) A Convergent 3-Block Semi-Proximal ADMM for Convex Minimization Problems with One Strongly Convex Block. Asia Pacific Journal of Operational Research, 32, 1550024.
[26] Candes, E.J. and Plan, Y. (2010) Matrix Completion with Noise. Proceedings of the IEEE, 98, 925-936.
[27] Zhang, Z. and Aeron, S. (2015) Exact Tensor Completion Using t-SVD.

comments powered by Disqus

Copyright © 2017 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.