Abstract
For the problem of low-rank tensor completion, rank estimation plays an extremely important role. And among some outstanding researches, nuclear norm is often used as a substitute of rank in the optimization due to its convex property. However, recent advances show that some non-convex functions could approximate the rank better, which can significantly improve the precision of the algorithm. While, the complexity of non-convex functions also leads to much higher computation cost, especially when the data are on a large scale. This paper proposes a mixture model for tensor completion by combining logDet function with Tucker decomposition, in which the logDet function is utilized as a much tighter rank approximation than the nuclear norm and the Tucker decomposition can significantly reduce the size of tensor that needs to be evaluated. In the implementation of the method, alternating direction method of multipliers is employed to obtain the optimal tensor completion. Several experiments are carried out to validate the effectiveness and efficiency of the method.
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De Lathauwer, L., Vandewalle, J.: Dimensionality reduction in higher-order signal processing and rank-(R1, R2, …, RN) reduction in multilinear algebra. Linear Algebra Appl. 391, 31–55 (2004)
Vlasic, D., Brand, M., Pfister, H., Popović, J.: Face transfer with multilinear models. ACM Trans. Graph. 24(3), 426–433 (2005)
Beylkin, G., Mohlenkamp, M.J.: Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. 99, 10246–10251 (2002)
Mørup, M.: Applications of tensor (multiway array) factorizations and decompositions in data mining. Wiley Interdiscip. Rev. Data Min. Knowl. Discov. 1, 124–140 (2011)
Komodakis, N., Tziritas, G.: Image completion using global optimization. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 442–449 (2006)
Patwardhan, K.A., Member, S.S., Sapiro, G., Member, S.S., Bertalmio, M.: Video inpainting under camera motion. IEEE Trans. Image Process. 16, 1–9 (2007)
Varghees, V.: Adaptive MRI image denoising using total-variation and local noise estimation. In: IEEE International Conference on Advances in Engineering Science and Management, pp. 506–511 (2012)
Li, N., Li, B.: Tensor completion for on-board compression of hyperspectral images. In: Proceedings of the International Conference on Image Processing (ICIP), pp. 517–520 (2010)
Filipovic, M., Jukic, A.: Tucker factorization with missing data with application to low-n-rank tensor completion. Multidimens. Syst. Signal Process. 26, 677–692 (2015)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)
Håstad, J.: Tensor rank is NP-complete. In: Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence (LNAI) and Lecture Notes in Bioinformatic), pp. 451–460 (1989)
Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471–501 (2010)
Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)
Liu, J., Musialski, P., Wonka, P., Ye, J.: Tensor completion for estimating missing values in visual data. IEEE Trans. Pattern Anal. Mach. Intell. 35 208–220 (2013)
Xu, Y., Hao, R., Yin, W., Su, Z.: Parallel matrix factorization for low-rank tensor completion. Inverse Probl. Imaging 9(2), 601–624 (2015)
Tomioka, R., Suzuki, T.: Convex tensor decomposition via structured schatten norm regularization. Adv. Neural Inf. Process. Syst. 1, 1331–1339 (2013)
Wimalawarne, K., Sugiyama, M., Tomioka, R.: Multitask learning meets tensor factorization: Task imputation via convex optimization. Adv. Neural Inf. Process. Syst. 4, 2825–2833 (2014)
Nimishakavi, M., Jawanpuria, P.K.: A dual framework for low-rank tensor completion. Adv. Neural Inf. Process. Syst. 31, 5484–5495 (2018)
Kang, Z., Peng, C., Cheng, Q.: Robust subspace clustering via smoothed rank approximation. IEEE Signal Process. Lett. 22, 2088–2092 (2015)
Ji, T.Y., Huang, T.Z., Le Zhao, X., Ma, T.H., Deng, L.J.: A non-convex tensor rank approximation for tensor completion. Appl. Math. Model. 48, 410–422 (2017)
Li, Y.F., Zhang, Y.J., Huang, Z.H.: A reweighted nuclear norm minimization algorithm for low rank matrix recovery. J. Comput. Appl. Math. 263, 338–350 (2014)
Gu, S., Zhang, L., Zuo, W., Feng, X.: Weighted nuclear norm minimization with application to image denoising. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 2862–2869 (2014)
Chen, S., Lyu, M.R., King, I., Xu, Z.: Exact and stable recovery of pairwise interaction tensors. Adv. Neural Inf. Process. Syst. 26, 1691–1699 (2013)
Zheng, Y.B., Huang, T.Z., Ji, T.Y., Zhao, X.L., Jiang, T.X.: Low-rank tensor completion via smooth matrix factorization. Appl. Math. Model. 70, 677–695 (2019)
Xu, Z., Yan, F., Qi, Y.: Bayesian nonparametric models for multiway data analysis. IEEE Trans. Pattern Anal. Mach. Intell. 37(2), 475–487 (2013)
Tucker, L.R.: Some mathematical notes on three-mode factor analysis. Psychometrika 31(3), 279–311 (1966)
Sheehan, B.N., Saad, Y.: Higher order orthogonal iteration of tensors (hooi) and its relation to pca and glram. In: Proceedings of the 2007 SIAM International Conference on Data Mining, pp. 355–366 (2007)
He, B., Tao, M., Yuan, X.: Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22(2), 313–340 (2012)
Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13, 600–612 (2004)
Zhang, Z., Ely, G., Aeron, S.: Novel methods for multilinear data completion and de-noising based on tensor-SVD. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3842–3849 (2014)
Oseledets, I.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)
Zhou, P., Lu, C., Member, S., Lin, Z., Member, S., Zhang, C.: Tensor factorization for low-rank tensor completion. IEEE Trans. Image Process. 27(3), 1152–1163 (2018)
Bengua, J.A., Phien, H.N., Tuan, H.D., Do, M.N.: Efficient tensor completion for color image and video recovery: low-rank tensor train. IEEE Trans. Image Process. 26(5), 2466–2479 (2017)
Acknowledgements
This research is supported by the National Key Research and Development Program of China (Project No. 2017YFD0700103) and National Natural Science Foundation of China (Grant Nos. #51475186 and #51775202).
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Shi, C., Huang, Z., Wan, L. et al. Low-rank tensor completion based on non-convex logDet function and Tucker decomposition. SIViP 15, 1169–1177 (2021). https://doi.org/10.1007/s11760-020-01845-7
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DOI: https://doi.org/10.1007/s11760-020-01845-7