Tensor completion via bilevel minimization with fixed-point constraint to estimate missing elements in noisy data


In this work, we consider the tensor completion problem of an incomplete and noisy observation. We introduce a novel completion model using bilevel minimization. Therefore, bilevel model-based denoising for the tensor completion problem is proposed. The denoising and completion tasks are fully separated. The upper-level directly addresses the completion problem with the truncated nuclear norm, while the lower-level uses the sparsity prior which is characterized by the l1-norm for the denoising task. Furthermore, we propose a simple strategy to solve our bilevel optimization problem. It formulates the lower-level as a fixed-point equation and then applies a simple but efficient iterative algorithm to get the reconstructed tensor. Numerically, the superiority of the proposal is reported via several experiments conducted on real data with an extremely small subset of observed entries.

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  1. 1.

    Candes, E.J., Recht, B.: Exact matrix completion via convex optimization. Foundations of Computational Mathematics 9(6), 717 (2009)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Lin, Z., Ganesh, A., Wright, J., Wu, L., Chen, M., Ma, Y.: Fast convex optimization algorithms for exact recovery of a corrupted low-rank matrix. UIUC Technical Report UILU-ENG-09-2214 (2009)

  3. 3.

    Banouar, O., Mohaoui, S., Raghay, S.: Collaborating filtering using unsupervised learning for image reconstruction from missing data. EURASIP Journal on Advances in Signal Processing 2018(1), 72 (2018)

    Article  Google Scholar 

  4. 4.

    Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction, vol. 5. SIAM, Philadelphia (2001)

    Google Scholar 

  5. 5.

    Unser, M., Chenouard, N.: A unifying parametric framework for 2D steerable wavelet transforms. SIAM J. Imag. Sci. 6(1), 102–135 (2013)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ochs, P., Ranftl, R., Brox, T., Pock, T.: Bilevel optimization with nonsmooth lower-level problems. In: International Conference on Scale Space and Variational Methods in Computer Vision, pp 654–665. Springer, Cham (2015)

  7. 7.

    Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM Journal on Imaging Sciences 6(2), 938–983 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Lu, C., Feng, J., Chen, Y., Liu, W., Lin, Z., Yan, S.: Tensor robust principal component analysis: exact recovery of corrupted low-rank tensors via convex optimization. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 5249–5257 (2016)

  9. 9.

    Xue, S., Qiu, W., Liu, F., Jin, X.: Low-rank tensor completion by truncated nuclear norm regularization. In: 2018 24th International Conference on Pattern Recognition (ICPR), pp 2600–2605. IEEE (2018)

  10. 10.

    Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22(4), 977–997 (1997)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dempe, S., Gadhi, N.: Necessary optimality conditions for bilevel set optimization problems. J. Glob. Optim. 39(4), 529–542 (2007)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Bard, J.F., Falk, J.E.: An explicit solution to the multi-level programming problem. Computers Operations Research 9(1), 77–100 (1982)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Ye, J.J.: Constraint qualifications and KKT conditions for bilevel programming problems. Math. Oper. Res. 31(4), 811–824 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Chen, G.H., Rockafellar, R.T.: Convergence rates in forward–backward splitting. SIAM J. Optim. 7(2), 421–444 (1997)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ma, S., Goldfarb, D., Chen, L.: Fixed-point and Bregman iterative methods for matrix rank minimization. Mathematical Programming 128(1-2), 321–353 (2011)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Zhang, D., Hu, Y., Ye, J., Li, X., He, X.: Matrix completion by truncated nuclear norm regularization. In: 2012 IEEE Conference on Computer Vision and Pattern Recognition, pp 2192–2199. Providence (2012)

  17. 17.

    Hu, Y., Zhang, D., Ye, J., Li, X., He, X.: Fast and accurate matrix completion via truncated nuclear norm regularization. In: IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, pp 2117–2130 (2013)

  18. 18.

    Hintermuller, M., Wu, T: Bilevel optimization for calibrating point spread functions in blind deconvolution (2015)

  19. 19.

    Song, Y., Li, J., Chen, X., Zhang, D., Tang, Q., Yang, K.: An efficient tensor completion method via truncated nuclear norm. J. Vis. Commun. Image Represent., 102791 (2020)

  20. 20.

    Rojo, O., Rojo, H.: Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices. Linear Algebra and its Applications 392, 211–233 (2004)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Liu, Q., Lai, Z., Zhou, Z., Kuang, F., Jin, Z.: A truncated nuclear norm regularization method based on weighted residual error for matrix completion. IEEE Trans. Image Process. 25(1), 316–330 (2015)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Yokota, T., Hontani, H.: Simultaneous visual data completion and denoising based on tensor rank and total variation minimization and its primal-dual splitting algorithm. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp 3732–3740 (2017)

  23. 23.

    Yokota, T., Hontani, H.: Simultaneous tensor completion and denoising by noise inequality constrained convex optimization. IEEE Access 7, 15669–15682 (2019)

    Article  Google Scholar 

  24. 24.

    Xu, R.H., Yin, W., Su, Z.: Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging 9 (2), 601–624 (2015). https://doi.org/10.3934/ipi.2015.9.601

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Souad Mohaoui.

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Mohaoui, S., Hakim, A. & Raghay, S. Tensor completion via bilevel minimization with fixed-point constraint to estimate missing elements in noisy data. Adv Comput Math 47, 10 (2021). https://doi.org/10.1007/s10444-020-09841-8

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  • Bilevel minimization
  • Tensor completion and denoising
  • Truncated nuclear norm
  • Fixed-point
  • Sparsity prior

Mathematics Subject Classification (2010)

  • 90C26
  • 68UXX
  • 65KXX