Abstract
Weighted nuclear norm provides a simple yet powerful tool to characterize the intrinsic low-rank structure of a matrix, and has been successfully applied to the matrix completion problem. However, in previous studies, the weighting functions to calculate the weights are fixed beforehand, and do not change during the whole iterative process. Such predefined weighting functions may not be able to precisely characterize the complicated structure underlying the observed data matrix, especially in the dynamic estimation process, and thus limits its performance. To address this issue, we propose a strategy of adaptive weighting function, for low-rank matrix/tensor completion. Specifically, we first parameterize the weighting function as a simple yet flexible neural network, that can approximate a wide range of monotonic decreasing functions. Then we propose an effective strategy, by virtue of the bi-level optimization technique, to adapt the weighting function, and incorporate this strategy to the alternating direction method of multipliers for solving low-rank matrix and tensor completion problems. Our empirical studies on a series of synthetic and real data have verified the effectiveness of the proposed approach, as compared with representative low-rank matrix and tensor completion methods.
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References
Krizhevsky A (2009) Learning multiple layers of features from tiny images. Technical report
Xiao T, Xia T, Yang Y, Huang C, Wang X (2015) Learning from massive noisy labeled data for image classification. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2691–2699
Felzenszwalb PF, Girshick RB, McAllester D, Ramanan D (2010) Object detection with discriminatively trained part-based models. IEEE Trans Pattern Anal Mach Intell 32(9):1627–1645
He K, Zhang X, Ren S, Sun J (2016) Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 770–778
Candès EJ, Recht B (2009) Exact matrix completion via convex optimization. Found Comput Math 9:717–772
Nie F, Huang H, Ding C (2012) Low-rank matrix recovery via efficient Schatten p-norm minimization. In: Proceedings of the 26th AAAI conference on artificial intelligence, pp 655–661
Marjanovic G, Solo V (2012) On \(l_q\) optimization and matrix completion. IEEE Trans Signal Process 60(11):5714–5724
Zhang D, Hu Y, Ye J, Li X, He X (2012) Matrix completion by truncated nuclear norm regularization. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2192–2199
Hu Y, Zhang D, Ye J, Li X, He X (2013) Fast and accurate matrix completion via truncated nuclear norm regularization. IEEE Trans Pattern Anal Mach Intell 35(9):2117–2130
Liu D, Zhou T, Qian H, Xu C, Zhang Z (2013) A nearly unbiased matrix completion approach. In: Proceedings of joint European conference on machine learning and knowledge discovery in databases, pp 210–225
Gui H, Han J, Gu Q (2016) Towards faster rates and oracle property for low-rank matrix estimation. In: Proceedings of the 33rd international conference on machine learning, pp 2300–2309
Fazel SM (2002) Matrix rank minimization with applications. PhD thesis, Stanford University
Gu S, Xie Q, Meng D, Zuo W, Feng X, Zhang L (2017) Weighted nuclear norm minimization and its applications to low level vision. Int J Comput Vis 121:183–208
Liu J, Musialski P, Wonka P, Ye J (2009) Tensor completion for estimating missing values in visual data. In: Proceedings of the 12th IEEE international conference on computer vision, pp 2114–2121
Liu J, Musialski P, Wonka P, Ye J (2013) Tensor completion for estimating missing values in visual data. IEEE Trans Pattern Anal Mach Intell 35(1):208–220
Nati NS, Jaakkola T (2003) Weighted low-rank approximations. In: Proceedings of the 20th international conference on machine learning, pp 720–727
Mnih A, Salakhutdinov RR (2007) Probabilistic matrix factorization. Adv Neural Inf Process Syst 20:1257–1264
Buchanan AM, Fitzgibbon AW (2005) Damped newton algorithms for matrix factorization with missing data. In: Proceedings of the IEEE conference on computer vision and pattern recognition, vol 2, pp 316–322
Okatani T, Deguchi K (2007) On the Wiberg algorithm for matrix factorization in the presence of missing components. Int J Comput Vis 72(3):329–337
Cai J-F, Candès EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982
Candes EJ, Plan Y (2010) Matrix completion with noise. Proc IEEE 98(6):925–936
Ma S, Goldfarb D, Chen L (2011) Fixed point and Bregman iterative methods for matrix rank minimization. Math Program 128:321–353
Mazumder R, Hastie T, Tibshirani R (2010) Spectral regularization algorithms for learning large incomplete matrices. J Mach Learn Res 11(80):2287–2322
Toh K-C, Yun S (2010) An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems. Pac J Optim 6(3):615–640
Lin Z, Chen M, Ma Y (2010) The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices. CoRR arXiv:1009.5055
Lin Z, Liu R, Su Z (2011) Linearized alternating direction method with adaptive penalty for low-rank representation. Adv Neural Inf Process Syst 24:612–620
Liu L, Huang W, Chen D-R (2014) Exact minimum rank approximation via Schatten p-norm minimization. J Comput Appl Math 267:218–227
Zhang R, Li S (2019) Optimal rip bounds for sparse signals recovery via \(\ell _p\) minimization. Appl Comput Harmon Anal 47(3):566–584
Zhang C-H (2010) Nearly unbiased variable selection under minimax concave penalty. Ann Stat 38(2):894–942
Fan J, Peng H (2004) Nonconcave penalized likelihood with a diverging number of parameters. Ann Stat 32(3):928–961
Zhang H, Qian J, Zhang B, Yang J, Gong C, Wei Y (2020) Low-rank matrix recovery via modified Schatten-\(p\) norm minimization with convergence guarantees. IEEE Trans Image Process 29:3132–3142
Lu C, Zhu C, Xu C, Yan S, Lin Z (2015) Generalized singular value thresholding. In: Proceedings of the twenty-ninth AAAI conference on artificial intelligence, pp 1805–1811
Lai M-J, Xu Y, Yin W (2013) Improved iteratively reweighted least squares for unconstrained smoothed \(\ell _q\) minimization. SIAM J Numer Anal 51(2):927–957
Li H, Lin Z (2015) Accelerated proximal gradient methods for nonconvex programming. Adv Neural Inf Process Syst 28:379–387
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends® Mach Learn 3(1):1–122
Kolda TG, Bader BW (2009) Tensor decompositions and applications. SIAM Rev 51(3):455–500
Romera-Paredes B, Pontil M (2013) A new convex relaxation for tensor completion. Adv Neural Inf Process Syst 26:2967–2975
Cao W, Wang Y, Yang C, Chang X, Han Z, Xu Z (2015) Folded-concave penalization approaches to tensor completion. Neurocomputing 152:261–273
Zhao Q, Meng D, Kong X, Xie Q, Cao W, Wang Y, Xu Z (2015) A novel sparsity measure for tensor recovery. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 271–279
Xie Q, Zhao Q, Meng D, Xu Z (2018) Kronecker-basis-representation based tensor sparsity and its applications to tensor recovery. IEEE Trans Pattern Anal Mach Intell 40(8):1888–1902
Xu Y, Hao R, Yin W, Su Z (2015) Parallel matrix factorization for low-rank tensor completion. Inverse Probl Imaging 9(2):601–624
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
Oseledets IV (2011) Tensor-train decomposition. SIAM J Sci Comput 33(5):2295–2317
Kilmer ME, Martin CD (2011) Factorization strategies for third-order tensors. Linear Algebra Appl 435(3):641–658
Kilmer ME, Braman K, Hao N, Hoover RC (2013) Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging. SIAM J Matrix Anal Appl 34(1):148–172
Martin CD, Shafer R, LaRue B (2013) An order-\$p\$ tensor factorization with applications in imaging. SIAM J Sci Comput 35(1):474–490
Zhang Z, Ely G, Aeron S, Hao N, Kilmer M (2014) Novel methods for multilinear data completion and de-noising based on tensor-SVD. In: Proceedings of IEEE conference on computer vision and pattern recognition, pp 3842–3849
Zhang Z, Aeron S (2017) Exact tensor completion using t-SVD. IEEE Trans Signal Process 65(6):1511–1526
Zheng Y-B, Huang T-Z, Zhao X-L, Jiang T-X, Ji T-Y, Ma T-H (2020) Tensor n-tubal rank and its convex relaxation for low-rank tensor recovery. Inf Sci 532:170–189
Liu X, Aeron S, Aggarwal V, Wang X (2020) Low-tubal-rank tensor completion using alternating minimization. IEEE Trans Inf Theory 66(3):1714–1737
Yuan L, Li C, Mandic DP, Cao J, Zhao Q (2019) Tensor ring decomposition with rank minimization on latent space: an efficient approach for tensor completion. In: Proceedings of the thirty-third AAAI conference on artificial intelligence, pp 9151–9158
Yuan L, Li C, Cao J, Zhao Q (2020) Rank minimization on tensor ring: an efficient approach for tensor decomposition and completion. Mach Learn 109(3):603–622
Wang W, Aggarwal V, Aeron S (2017) Efficient low rank tensor ring completion. In: Proceedings of the 2017 IEEE international conference on computer vision, pp 5698–5706
Yu J, Zhou G, Li C, Zhao Q, Xie S (2021) Low tensor-ring rank completion by parallel matrix factorization. IEEE Trans Neural Netw Learn Syst 32(7):3020–3033
Zhao Q, Zhou G, Xie S, Zhang L, Cichocki A (2016) Tensor ring decomposition. CoRR arXiv:1606.05535
Gu S, Zhang L, Zuo W, Feng X (2014) Weighted nuclear norm minimization with application to image denoising. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2862–2869
Lange K (2016) MM optimization algorithms. Society for Industrial and Applied Mathematics, Philadelphia
Candès EJ, Wakin MB, Boyd SP (2008) Enhancing sparsity by reweighted \(\ell _1\) minimization. J Fourier Anal Appl 14(5):877–905
Cybenko G (1989) Approximation by superpositions of a sigmoidal function. Math Control Signals Syst 2(4):303–314
Hornik K, Stinchcombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Netw 2(5):359–366
Barron AR (1993) Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans Inf Theory 39(3):930–945
Daniels H, Velikova M (2010) Monotone and partially monotone neural networks. IEEE Trans Neural Netw 21(6):906–917
Dugas C, Bengio Y, Bélisle F, Nadeau C, Garcia R (2009) Incorporating functional knowledge in neural networks. J Mach Learn Res 10(42):1239–1262
Finn C, Abbeel P, Levine S (2017) Model-agnostic meta-learning for fast adaptation of deep networks. In: Proceedings of the 34th international conference on machine learning, pp 1126–1135
Ren M, Zeng W, Yang B, Urtasun R (2018) Learning to reweight examples for robust deep learning. In: Proceedings of the 35th international conference on machine learning, pp 4334–4343
Shu J, Xie Q, Yi L, Zhao Q, Zhou S, Xu Z, Meng D (2019) Meta-weight-net: learning an explicit mapping for sample weighting. Adv Neural Inf Process Syst 32:1919–1930
Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L, Desmaison A, Kopf A, Yang E, DeVito Z, Raison M, Tejani A, Chilamkurthy S, Steiner B, Fang L, Bai J, Chintala S (2019) Pytorch: an imperative style, high-performance deep learning library. Adv Neural Inf Process Syst 32:8024–8035
Abadi M, Barham P, Chen J, Chen Z, Davis A, Dean J, Devin M, Ghemawat S, Irving G, Isard M, Kudlur M, Levenberg J, Monga R, Moore S, Murray D.G, Steiner B, Tucker P, Vasudevan V, Warden P, Wicke M, Yu Y, Zheng X (2016) Tensorflow: a system for large-scale machine learning. In: Proceedings of the 12th USENIX conference on operating systems design and implementation, pp 265–283
Xu Z, Chang X, Xu F, Zhang H (2012) \(l_{1/2}\) regularization: a thresholding representation theory and a fast solver. IEEE Trans Neural Netw Learn Syst 23(7):1013–1027
Wang Z, Bovik AC, Sheikh HR, Simoncelli EP (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13(4):600–612
Roth S, Black MJ (2009) Fields of experts. Int J Comput Vis 82(2):205–229
Zhang L, Song L, Du B, Zhang Y (2021) Nonlocal low-rank tensor completion for visual data. IEEE Trans Cybern 51(2):673–685
Yasuma F, Mitsunaga T, Iso D, Nayar SK (2010) Generalized assorted pixel camera: postcapture control of resolution, dynamic range, and spectrum. IEEE Trans Image Process 19(9):2241–2253. https://doi.org/10.1109/TIP.2010.2046811
Candès EJ, Li X, Ma Y, Wright J (2011) Robust principal component analysis? J ACM 58(3):11
Acknowledgements
This work was supported National Key Research and Development Program of China (2020YFA0713900), China NSFC Projects (62076196) and the Macao Science and Technology Development Fund (061/2020/A2).
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Zhao, Q., Lin, Y., Wang, F. et al. Adaptive weighting function for weighted nuclear norm based matrix/tensor completion. Int. J. Mach. Learn. & Cyber. 15, 697–718 (2024). https://doi.org/10.1007/s13042-023-01935-1
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DOI: https://doi.org/10.1007/s13042-023-01935-1