Abstract
Low-rank tensor completion and recovery have received considerable attention in the recent literature. The existing algorithms, however, are prone to suffer a failure when the multiway data are simultaneously contaminated by arbitrary outliers and missing values. In this paper, we study the unsupervised tensor learning problem, in which a low-rank tensor is recovered from an incomplete and grossly corrupted multidimensional array. We introduce a unified framework for this problem by using a simple equation to replace the linear projection operator constraint, and further reformulate it as two convex optimization problems through different approximations of the tensor rank. Two globally convergent algorithms, derived from the alternating direction augmented Lagrangian (ADAL) and linearized proximal ADAL methods, respectively, are proposed for solving these problems. Experimental results on synthetic and real-world data validate the effectiveness and superiority of our methods.
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References
Acar E, Dunlavy DM, Kolda TG, Mørup M (2010) Scalable tensor factorizations with missing data. In: SIAM international conference on data mining, pp 701–712
Cai JF, CandèS EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982
Candès EJ, Li X, Ma Y, Wright J (2011) Robust principal component analysis? J ACM 58(3):1–37
Chen Y, Jalali A, Sanghavi S, Caramanis C (2013) Low-rank matrix recovery from errors and erasures. IEEE Trans Inf Theory 59(7):4324–4337
De Lathauwer L, Vandewalle J (2004) Dimensionality reduction in higher-order signal processing and rank-(\(r_{1}\),\(r_{2}\), \(\cdots\), \(r_{N}\)) reduction in multilinear algebra. Linear Algebra Appl 391(1):31–55
Eckstein J, Bertsekas DP (1992) On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math Program 55(1–3):293–318
Filipović M, Jukić A (2015) Tucker factorization with missing data with application to low-\(n\)-rank tensor completion. Multidimens Syst Signal Process 26(3):1–16
Gandy S, Recht B, Yamada I (2011) Tensor completion and low-n-rank tensor recovery via convex optimization. Inverse Probl 27(2):025010
Goldfarb D, Qin Z (2014) Robust low-rank tensor recovery: models and algorithms. SIAM J Matrix Anal Appl 35(1):225–253
Harshman RA (2009) Foundations of the PARAFAC procedure: model and conditions for an “explanatory” multi-mode factor analysis. UCLA Work Pap Phonetic 16:1–84
Huang B, Mu C, Goldfarb D, Wright J (2014) Provable low-rank tensor recovery. In: Optimization Online, p 4252
Kolda TG, Sun J (2008) Scalable tensor decompositions for multi-aspect data mining. In: 8th IEEE international conference on data mining, pp 363–372
Lee KC, Ho J, Kriegman DJ (2005) Acquiring linear subspaces for face recognition under variable lighting. IEEE Trans Pattern Anal Mach Intell 27(5):684–698
Li X (2013) Compressed sensing and matrix completion with constant proportion of corruptions. Constr Approx 37(1):73–99
Li Y, Yan J, Zhou Y, Yang J (2010) Optimum subspace learning and error correction for tensors. In: European conference on computer vision conference on computer vision, pp 790–803
Liu J, Musialski P, Wonka P, Ye J (2009) Tensor completion for estimating missing values in visual data. In: 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
Mu C, Huang B, Wright J, Goldfarb D (2013) Square deal: lower bounds and improved relaxations for tensor recovery. In: International conference on machine learning, pp 73–81
Recht B (2011) A simpler approach to matrix completion. J Mach Learn Res 12(4):3413–3430
Shang F, Liu Y, Cheng J, Cheng H (2014) Robust principal component analysis with missing data. In: Proceedings of the 23rd ACM international conference on information and knowledge management, Shanghai, China, pp 1149–1158
Signoretto M, Dinh QT, Lathauwer LD, Suykens JAK (2014) Learning with tensors: a framework based on convex optimization and spectral regularization. Mach Learn 94(3):303–351
Signoretto M, Lathauwer LD, Suykens JAK (2010) Nuclear norms for tensors and their use for convex multilinear estimation. Technical Report 10-186, ESAT-SISTA, K. U. Leuven, Leuven, Belgium
Tan H, Cheng B, Feng J, Feng G, Wang W, Zhang Y-J (2013) Low-\(n\)-rank tensor recovery based on multi-linear augmented lagrange multiplier method. Neurocomputing 119:144–152
Tomioka R, Hayashi K, Kashima H (2010) Estimation of low-rank tensors via convex optimization. arXiv:1010.0789
Tomioka R, Suzuki T, Hayashi K, Kashima H (2011) Statistical performance of convex tensor decomposition. In: Advances in neural information processing systems, pp 972–980
Tucker LR (1966) Some mathematical notes on three-mode factor analysis. Psychometrika 31(3):279–311
Yang J, Yuan X (2013) Linearized augmented lagrangian and alternating direction methods for nuclear norm minimization. Math Comput 82(281):301–329
Yang J, Zhang Y (2011) Alternating direction algorithms for \(ell _1\)-problems in compressive sensing. SIAM J Sci Comput 35(1):250–278
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This work was supported by National Natural Science Foundation of China Nos. 61702023 and 91538204.
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Meng, Z., Zhou, Y. & Zhao, Y. Unsupervised learning low-rank tensor from incomplete and grossly corrupted data. Neural Comput & Applic 31, 8327–8335 (2019). https://doi.org/10.1007/s00521-018-3899-x
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DOI: https://doi.org/10.1007/s00521-018-3899-x