Abstract
Low-rank tensor completion aims to recover the missing entries of the tensor from its partially observed data by using the low-rank property of the tensor. Since rank minimization is an NP-hard problem, the convex surrogate nuclear norm is usually used to replace the rank norm and has obtained promising results. However, the nuclear norm is not a tight envelope of the rank norm and usually over-penalizes large singular values. In this paper, inspired by the effectiveness of the matrix Schatten-q norm, which is a tighter approximation of rank norm when 0 < q < 1, we generalize the matrix Schatten-q norm to tensor case and propose a Unitary Transformed Tensor Schatten-q Norm (UTT-Sq) with an arbitrary unitary transform matrix. More importantly, the factor tensor norm surrogate theorem is derived. We prove large-scale UTT-Sq norm (which is nonconvex and not tractable when 0 < q < 1) is equivalent to minimizing the weighted sum formulation of multiple small-scale UTT-\(S_{q_{i}}\) (with different qi and qi ≥ 1). Based on this equivalence, we propose a low-rank tensor completion framework using Unitary Transformed Tensor Multi-Factor Norm (UTTMFN) penalty. The optimization problem is solved using the Alternating Direction Method of Multipliers (ADMM) with the proof of convergence. Experimental results on synthetic data, images and videos show that the proposed UTTMFN can achieve competitive results with the state-of-the-art methods for tensor completion.
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Data Availability
The data supporting the study’s findings are available from the corresponding author, lianyi_1999@nuaa.edu.cn or jltian@nuaa.edu.cn, upon reasonable request.
References
Song G, Ng MK, Zhang X (2020) Robust tensor completion using transformed tensor singular value decomposition. Numer Linear Alg Appl 27(3):2299. https://doi.org/10.1002/nla.2299
Ji T-Y, Huang T-Z, Zhao X-L, Ma T-H, Deng L-J (2017) A non-convex tensor rank approximation for tensor completion. Appl Math Model 48:410–422. https://doi.org/10.1016/j.apm.2017.04.002
Hosono K, Ono S, Miyata T (2019) Weighted tensor nuclear norm minimization for color image restoration. IEEE Access 7:88768–88776. https://doi.org/10.1109/ACCESS.2019.2926507
Du S, Liu B, Shan G, Shi Y, Wang W (2022) Enhanced tensor low-rank representation for clustering and denoising. Knowl-Based Syst 243:108468. https://doi.org/10.1016/j.knosys.2022.108468
Zhang H, Qian J, Zhang B, Yang J, Gong C, Wei Y (2019) Low-rank matrix recovery via modified schatten-p norm minimization with convergence guarantees. IEEE Trans Image Process 29:3132–3142. https://doi.org/10.1109/TIP.2019.2957925
Hu Z, Nie F, Wang R, Li X (2021) Low rank regularization: a review. Neural Netw 136:218–232. https://doi.org/10.1016/j.neunet.2020.09.021
Shang F, Cheng J, Liu Y, Luo Z-Q, Lin Z (2017) Bilinear factor matrix norm minimization for robust pca: Algorithms and applications. IEEE Trans Pattern Anal Mach Intell 40(9):2066–2080. https://doi.org/10.1109/TPAMI.2017.2748590
Shi C, Huang Z, Wan L, Xiong T (2021) Low-rank tensor completion based on non-convex logdet function and tucker decomposition. Signal Image Video Process 15(6):1169–1177. https://doi.org/10.1007/s11760-020-01845-7
Kong H, Xie X, Lin Z (2018) t-schatten-p norm for low-rank tensor recovery. IEEE J Sel Top Signal Process 12 (6):1405–1419. https://doi.org/10.1109/JSTSP.2018.2879185
Hitchcock FL (1927) The expression of a tensor or a polyadic as a sum of products. J Math Phys 6(1-4):164–189. https://doi.org/10.1002/sapm192761164
Tucker LR (1966) Some mathematical notes on three-mode factor analysis. Psychometrika 31 (3):279–311. https://doi.org/10.1007/BF02289464
Zhao Q, Zhou G, Xie S, Zhang L, Cichocki A (2016) Tensor ring decomposition. Preprint at https://doi.org/10.48550/arXiv.1606.05535
Bengua JA, Phien HN, Tuan HD, Do MN (2017) Efficient tensor completion for color image and video recovery: low-rank tensor train. IEEE Trans Image Process 26(5):2466–2479. https://doi.org/10.1109/TIP.2017.2672439
Zhang Z, Ely G, Aeron S, Hao N, Kilmer M (2014) Novel methods for multilinear data completion and de-noising based on tensor-svd. :3842–3849
Carroll JD, Chang J-J (1970) Analysis of individual differences in multidimensional scaling via an n-way generalization of “eckart-young” decomposition. Psychometrika 35(3):283–319. https://doi.org/10.1007/BF02310791
Kilmer ME, Martin CD (2011) Factorization strategies for third-order tensors. Linear Alg Appl 435(3):641–658. https://doi.org/10.1016/j.laa.2010.09.020
Acar E, Dunlavy DM, Kolda TG, Mørup M (2011) Scalable tensor factorizations for incomplete data. Chemom Intell Lab Syst 106(1):41–56. https://doi.org/10.1016/j.chemolab.2010.08.004
Liu J, Musialski P, Wonka P, Ye J (2012) Tensor completion for estimating missing values in visual data. IEEE Trans Pattern Anal Mach Intell 35(1):208–220. https://doi.org/10.1109/TPAMI.2012.39
Yu J, Zhou G, Li C, Zhao Q, Xie S (2020) Low tensor-ring rank completion by parallel matrix factorization. IEEE Trans Neural Netw Learn Syst 32(7):3020–3033. https://doi.org/10.1109/TNNLS.2020.3009210
He J, Zheng X, Gao P, Zhou Y (2022) Low-rank tensor completion based on tensor train rank with partially overlapped sub-blocks. Signal Process 190:108339. https://doi.org/10.1016/j.sigpro.2021.108339
Kong H, Lu C, Lin Z (2021) Tensor q-rank: New data dependent definition of tensor rank. Mach Learn 110(7):1867–1900. https://doi.org/10.1007/s10994-021-05987-8
Lu C, Peng X, Wei Y (2019) Low-rank tensor completion with a new tensor nuclear norm induced by invertible linear transforms. :5996–6004
Kernfeld E, Kilmer M, Aeron S (2015) Tensor–tensor products with invertible linear transforms. Linear Alg Appl 485:545–570. https://doi.org/10.1016/j.laa.2015.07.021
Li B-Z, Zhao X-L, Ji T-Y, Zhang X-J, Huang T-Z (2021) Nonlinear transform induced tensor nuclear norm for tensor completion. arXiv:2110.08774. Preprint at https://doi.org/10.48550/arXiv.2110.08774
Zhou P, Lu C, Lin Z, Zhang C (2017) Tensor factorization for low-rank tensor completion. IEEE Trans Image Process 27(3):1152–1163. https://doi.org/10.1109/TIP.2017.2762595
Xu Y (2017) Fast algorithms for higher-order singular value decomposition from incomplete data. J Comput Math 35(4):397–422. https://doi.org/10.4208/jcm.1608-m2016-0641
Du S, Xiao Q, Shi Y, Cucchiara R, Ma Y (2021) Unifying tensor factorization and tensor nuclear norm approaches for low-rank tensor completion. Neurocomputing 458:204–218. https://doi.org/10.1016/j.neucom.2021.06.020
Jiang T-X, Huang T-Z, Zhao X-L, Deng L-J (2020) Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm. J Comput Appl Math 372:112680. https://doi.org/10.1016/j.cam.2019.112680
Xu W-H, Zhao X-L, Ji T-Y, Miao J-Q, Ma T-H, Wang S, Huang T-Z (2019) Laplace function based nonconvex surrogate for low-rank tensor completion. Signal Process Image Commun 73:62–69. https://doi.org/10.1016/j.image.2018.11.007
Shi C, Huang Z, Wan L, Xiong T (2019) Low-rank tensor completion based on log-det rank approximation and matrix factorization. J Sci Comput 80(3):1888–1912. https://doi.org/10.1007/s10915-019-01009-x
Wang H, Zhang F, Wang J, Huang T, Huang J, Liu X (2021) Generalized nonconvex approach for low-tubal-rank tensor recovery. IEEE Transactions on Neural Networks and Learning Systems. https://doi.org/10.1109/TNNLS.2021.3051650
Lu C, Feng J, Chen Y, Liu W, Lin Z, Yan S (2016) Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization. :5249–5257
Trzasko J, Manduca A (2008) Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization. IEEE Trans Med Imaging 28 (1):106–121. https://doi.org/10.1109/TMI.2008.927346
Fazel M, Hindi H, Boyd SP (2003) Log-det heuristic for matrix rank minimization with applications to hankel and euclidean distance matrices. 3:2156–2162. https://doi.org/10.1109/ACC.2003.1243393
Gao C, Wang N, Yu Q, Zhang Z (2011) A feasible nonconvex relaxation approach to feature selection. 25(1):356–361. https://ojs.aaai.org/index.php/AAAI/article/view/7921
Gu S, Zhang L, Zuo W, Feng X (2014) Weighted nuclear norm minimization with application to image denoising. :2862–2869
Shang F, Liu Y, Shang F, Liu H, Kong L, Jiao L (2020) A unified scalable equivalent formulation for schatten quasi-norms. Mathematics 8(8):1325. https://doi.org/10.3390/math8081325
Wang F, Cao W, Xu Z (2018) Convergence of multi-block bregman admm for nonconvex composite problems. Sci China Inf Sci 61(12):1–12. https://doi.org/10.1007/s11432-017-9367-6
Epton MA (1980) Methods for the solution of axd- bxc= e and its application in the numerical solution of implicit ordinary differential equations. BIT Numer Math 20(3):341–345. https://doi.org/10.1007/BF01932775
Cai J-F, Candès EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982. https://doi.org/10.1137/080738970
Zuo W, Meng D, Zhang L, Feng X, Zhang D (2013) A generalized iterated shrinkage algorithm for non-convex sparse coding. :217–224
Chen C, He B, Ye Y, Yuan X (2016) The direct extension of admm for multi-block convex minimization problems is not necessarily convergent. Math Program 155(1):57–79. https://doi.org/10.1007/s10107-014-0826-5
Jiang T-X, Zhao X-L, Zhang H, Ng MK (2021) Dictionary learning with low-rank coding coefficients for tensor completion. IEEE Transactions on Neural Networks and Learning Systems. https://doi.org/10.1109/TNNLS.2021.3104837
Chen L, Jiang X, Liu X, Zhou Z (2021) Logarithmic norm regularized low-rank factorization for matrix and tensor completion. IEEE Trans Image Process 30:3434–3449. https://doi.org/10.1109/TIP.2021.3061908
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Tian, J., Zhu, Y. & Liu, J. A general multi-factor norm based low-rank tensor completion framework. Appl Intell 53, 19317–19337 (2023). https://doi.org/10.1007/s10489-023-04477-9
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DOI: https://doi.org/10.1007/s10489-023-04477-9