In recent years, there are many surrogates for tensor tubal rank. In this paper, we propose a hybrid norm consisting of the weighted nuclear norm and the weighted Frobenius norm (WTNFN) of a tensor. The WTNFN is a surrogate for tensor tubal rank, and studies the weighted tensor nuclear and Frobenius norm minimization (WTNFNM) problem. The aim is to enhance the stability of the solution and improve the shortcomings of the traditional method of minimizing approximate rank functions based on the tensor nuclear norm (TNN) in the field of low-rank tensor recovery. Based on this definition, we build a novel model for typical tensor recovery problems i.e. the weighted tensor nuclear and Frobenius tensor completion (WTNFNTC). The experimental results on synthetic data and real data show that the stability and recovery effects of the model are improved compared with related algorithms.
Cite this paper
Wang, X. and Jiang, W. (2022). Improved Robust Low-Rank Regularization Tensor Completion. Open Access Library Journal, 9, e9425. doi: http://dx.doi.org/10.4236/oalib.1109425.
Tucker, L.R. (1963) Implications of Factor Analysis of Three-Way Matrices for Measurement of Change. Problems in Measuring Change, 15, 122-137.
https://doi.org/10.1108/09534810210423008
Vasilescu, M.A.O. and Terzopoulos, D. (2002) Multilinear Analysis of Image Ensembles: TensorFaces. In: Heyden, A., et al., Eds., European Conference on Computer Vision, Springer, Berlin, 447-460. https://doi.org/10.1007/3-540-47969-4_30
Sidiropoulos, N.D., De Lathauwer, L., Fu, X., et al. (2017) Tensor Decomposition for Signal Processing and Machine Learning. IEEE Transactions on Signal Processing, 65, 3551-3582. https://doi.org/10.1109/TSP.2017.2690524
Lai, Z., Wong, W.K., Jin, Z., et al. (2012) Sparse Approximation to the Eigensubspace for Discrimination. IEEE Transactions on Neural Networks and Learning Systems, 23, 1948-1960. https://doi.org/10.1109/TNNLS.2012.2217154
De Lathauwer, L. and De Moor, B. (1998) From Matrix to Tensor: Multilinear Algebra and Signal Processing. Institute of Mathematics and Its Applications Conference Series 67, Oxford University Press, Oxford, 1-16.
Cichocki, A., Mandic, D., De Lathauwer, L., et al. (2015) Tensor Decompositions for Signal Processing Applications: From Two-Way to Multiway Component Analysis. IEEE Signal Processing Magazine, 32, 145-163.
https://doi.org/10.1109/MSP.2013.2297439
Song, Q., Huang, X., Ge, H., et al. (2017) Multi-Aspect Streaming Tensor Completion. Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Halifax, 13-17 August 2017, 435-443.
https://doi.org/10.1145/3097983.3098007
Kiers, H.A. (2000) Towards a Standardized Notation and Terminology in Multiway Analysis. Journal of Chemometrics: A Journal of the Chemometrics Society, 14, 105-122.
https://doi.org/10.1002/1099-128X(200005/06)14:3<105::AID-CEM582>3.0.CO;2-I
Kapteyn, A., Neudecker, H. and Wansbeek, T. (1986) An Approach Ton-Mode Components Analysis. Psychometrika, 51, 269-275.
https://doi.org/10.1007/BF02293984
Liu, J., Musialski, P., Wonka, P. and Ye, J. (2013) Tensor Completion for Estimating Missing Values in Visual Data. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35, 208-220. https://doi.org/10.1109/TPAMI.2012.39
Romera-Paredes, B. and Pontil, M. (2013) A New Convex Relaxation for Tensor Completion. 27th Annual Conference on Neural Information Processing Systems, Lake Tahoe, 5-10 December 2013, 2967-2975.
Kilmer, M.E. and Martin, C.D. (2011) Factorization Strategies for Third-Order Tensors. Linear Algebra and Its Applications, 435, 641-658.
https://doi.org/10.1016/j.laa.2010.09.020
Lu, C., Feng, J., Liu, W., Lin, Z. and Yan, S. (2020) Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42, 925-938.
Peng, X., Lu, C., Yi, Z. and Tang, H. (2018) Connections between Nuclear-Norm and Frobenius-Norm-Based Representations. IEEE Transactions on Neural Networks and Learning Systems, 29, 218-224. https://doi.org/10.1109/TNNLS.2016.2608834
Zhang, H., Yi, Z. and Peng, X. (2014) fLRR: Fast Low-Rank Representation Using Frobenius-Norm. Electronics Letters, 50, 936-938.
https://doi.org/10.1049/el.2014.1396
Kilmer, M.E., Braman, K., Hao, N. and Hoover, R.C. (2013) Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging. SIAM Journal on Matrix Analysis and Applications, 34, 148-172.
https://doi.org/10.1137/110837711
Zhang, Z., Ely, G., Aeron, S., Hao, N. and Kilmer, M. (2014) Novel Methods for Multilinear Data Completion and De-Noising Based on Tensor-SVD. 2014 IEEE Conference on Computer Vision and Pattern Recognition, Columbus, 23-28 June 2014, 3842-3849.
Gu, S., Xie, Q., Meng, D., et al. (2017) Weighted Nuclear Norm Minimization and Its Applications to Low Level Vision. International Journal of Computer Vision, 121, 183-208. https://doi.org/10.1007/s11263-016-0930-5
Clarke, F.H. (1990) Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics, Philadelphia. https://doi.org/10.1137/1.9781611971309
Beck, A. and Teboulle, M. (2009) A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2, 183-202.
https://doi.org/10.1137/080716542
Lu, C., Peng, X. and Wei, Y. (2019) Low-Rank Tensor Completion with a New Tensor Nuclear Norm Induced by Invertible Linear Transforms. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, Long Beach, 16-20 June 2019, 5996-6004. https://doi.org/10.1109/CVPR.2019.00615
