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Recovery of Corrupted Low-Rank Tensors
Department of Electronic Science and Engineering, National University of Defense Technology, Changsha, China
Department of Electronic Science and Engineering, National University of Defense Technology, Changsha, China
- 1 Department of Electronic Science and Engineering, National University of Defense Technology, Changsha, China
- 2 Department of Electronic Science and Engineering, National University of Defense Technology, Changsha, China
Applied Mathematics·Volume 08 (2017)·Pages 229–244·Published 7 February 2017·DOI10.4236/am.2017.82019
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Abstract
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 l 1 -norm in a unified convex relaxation framework. The nuclear norm is adopted to explore the low-rank components and the l 1 -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.
KeywordsLow-Rank TensorTensor RecoveryAugmented Lagrangian MethodImpulsive NoiseMixed Noise
- Jolliffe, I. (2002) Principal Component Analysis. Wiley Online Library.
- Candes, E.J., Li, X.D., Ma, Y. and Wright, J. (2011) Robust Principal Component Analysis? Journal of the ACM, 58, Article Number: 11. https://doi.org/10.1145/1970392.1970395
- 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. https://doi.org/10.1137/090761793
- 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).
- 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. https://doi.org/10.1137/100781894
- Candes, E.J. and Recht, B. (2009) Exact Matrix Completion via Convex Optimization. Foundations of Computational Mathematics, 9, 717-772. https://doi.org/10.1007/s10208-009-9045-5
- Candes, E.J. and Tao, T. (2010) The Power of Convex Relaxation: Near-Optimal Matrix Completion. IEEE Transaction on Information Theory, 56, 2053-2080. https://doi.org/10.1109/TIT.2010.2044061
- Wang, H. and Ahuja, N. (2004) Compact Representation of Multidimensional Data Using Tensor Rank-One Decomposition.
- 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.
- 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.
- 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. https://doi.org/10.1137/090755138
- 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. https://doi.org/10.1063/1.3451264
- Hu, S. and Qi, L. (2012) Algebraic Connectivity of an Even Uniform Hypergraph. Journal of Combinatorial Optimization, 24, 564-579. https://doi.org/10.1007/s10878-011-9407-1