Skip to main content
SpringerLink
Log in

Robust Schatten-p Norm Based Approach for Tensor Completion

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The matrix nuclear norm has been widely applied to approximate the matrix rank for low-rank tensor completion because of its convexity. However, this relaxation may make the solution seriously deviate from the original solution for real-world data recovery. In this paper, using a nonconvex approximation of rank, i.e., the Schatten-p norm, we propose a novel model for tensor completion. It’s hard to solve this model directly because the objective function of the model is nonconvex. To solve the model, we develop a variant of this model via the classical quadric penalty method, and propose an algorithm, i.e., SpBCD, based on the block coordinate descent method. Although the objective function of the variant is nonconvex, we show that the sequence generated by SpBCD is convergent to a critical point. Our numerical experiments on real-world data show that SpBCD delivers state-of-art performance in recovering missing data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. http://www1.cs.columbia.edu/CAVE/databases/multispectral/.

  2. http://www.osirix-viewer.com/resources/dicom-image-library/.

  3. http://trace.eas.asu.edu/yuv/.

References

  1. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of ACM Siggraph, pp. 414–424, New Orleans, USA (2000)

  2. Komodakis, N., Tziritas, G.: Image completion using global optimization. In: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, pp. 417–424 (2006)

  3. Korah, T., Rasmussen, C.: Spatio-temporal impainting for recovering texture maps of occluded building facades. IEEE Trans. Image Process. 16(7), 2262–2271 (2007)

    Article  MathSciNet  Google Scholar 

  4. Patwardhan, K.A., Spiro, G., Bertalmio, M.: Video inpainting under constrained camera motion. IEEE Trans. Image Process. 16(2), 545–553 (2007)

    Article  MathSciNet  Google Scholar 

  5. Varghees, V.N., Manikandan, M.S., Gini, R.: Adaptive MRI image denoising using total-variation and local noise estimation. In: Processing of the 2012 International Conference on Advances in Engineering, Science and Management (ICAESM), pp. 506–511 (2012)

  6. Li, N., Li, B.: Tensor completion for on-board compression of hyperspectral images. In: IEEE International Conference on Image Processing, pp. 517–520 (2015)

  7. Xing, Z., Zhou, M., Castrodad, A., Saprio, G., Carin, L.: Dictionary learning for noisy and incomplete hyperspectral images. SIAM J. Imaging Sci. 5(1), 33–56 (2012)

    Article  MathSciNet  Google Scholar 

  8. Liu, Y., Jiao, L., Shang, F.: An efficient matrix factorization method for tensor completion. IEEE Signal Process. Lett. 20(4), 307–310 (2013)

    Article  Google Scholar 

  9. 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)

    Article  MathSciNet  Google Scholar 

  10. Nie, F., Wang, H., Cai, X., Huang, H., Ding, C.: Robust matrix completion via joint schatten-p norm and $ L_p $ norm minimization. In: ICDM, pp. 566–574 (2012)

  11. Cai, T.T., Zhou, W.: Matrix completion via max-norm constrained optimization. Electron. J. Stat. 10(1), 1493–1525 (2016)

    Article  MathSciNet  Google Scholar 

  12. Koltchinskii, Vladimir: Von neumann entropy penalization and low-rank matrix estimation. Ann. Stat. 39(6), 2936–2973 (2011)

    Article  MathSciNet  Google Scholar 

  13. Ma, S., Goldfarb, D., Chen, L.: Fixed point and bregman iterative methods for matrix rank minimization. Math. Program. 128(1–2), 321–353 (2011)

    Article  MathSciNet  Google Scholar 

  14. Toh, K.C., Tun, S.: An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems. Pac. J. Optim. 6(3), 615–640 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Wen, Z., Yin, W., Zhang, Y.: Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm. Math. Program. Comput. 4(4), 1–29 (2012)

    Article  MathSciNet  Google Scholar 

  16. Chen, C., He, B., Yuan, X.: Matrix completion via an alternating direction method. IMA J. Numer. Anal. 32(1), 227–245 (2012)

    Article  MathSciNet  Google Scholar 

  17. Xu, Y., Yin, W., Wen, Z., Zhang, Y.: An alternating direction algorithm for matrix completion with nonnegative factors. J. Front. Math. China 7(2), 365–384 (2011)

    Article  MathSciNet  Google Scholar 

  18. Tucker, L.R.: Implications of factor analysis of three-way matrices for measurement of change. In: Harris, C.W. (ed.) Problems in Measuring Change, pp. 122–137. University of Wisconsin Press, Madison (1963)

    Google Scholar 

  19. Tucker, L.R.: Some mathematical notes on three-mode factor analysis. Psychometrika 31(3), 279–311 (1966)

    Article  MathSciNet  Google Scholar 

  20. Hackbusch, W., Küuhn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15(5), 706–722 (2009)

    Article  MathSciNet  Google Scholar 

  21. Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31(4), 2029–2054 (2010)

    Article  MathSciNet  Google Scholar 

  22. Oseledets, I.V., Tyrtyshnikov, E.E.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31(5), 3744–3759 (2009)

    Article  MathSciNet  Google Scholar 

  23. Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)

    Article  MathSciNet  Google Scholar 

  24. Kruskal, J.B.: Rank decomposition and uniqueness for 3-way and N-way arrays. In: Multiway Data Analysis, pp. 7–18 (1988)

  25. Liu, J., Musialski, P., Wonka, P., Ye, J.P.: Tensor completion for estimating missing values in visual data. IEEE Trans. Pattern Anal. Mach. Intell. 35(1), 1126–1153 (2013)

    Google Scholar 

  26. Kolda, G., Bader, W.: Tensor decomposition and application. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  Google Scholar 

  27. Kressner, D., Steinlechner, M., Vandereycken, B.: Low-rank tensor completion by Riemannian optimization. BIT Numer. Math. 54(2), 447–468 (2014)

    Article  MathSciNet  Google Scholar 

  28. Xu, Y., Hao, R., Yin, W., Su, Z.: Parallel matrix factorization for low-rank tensor completion. Inverse Probl. Imaging 9(2), 601–624 (2015)

    Article  MathSciNet  Google Scholar 

  29. Gao, S., Fan, Q.: A mixture of nuclear norm and matrix factorization for tensor completion. J. Sci. Comput. 75(1), 43–64 (2018)

    Article  MathSciNet  Google Scholar 

  30. Sun, W., Yuan, Y.: Optimization Theory and Methods: Nonlinear Programming. Springer, New York (2006)

    MATH  Google Scholar 

  31. Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9(6), 717–772 (2009)

    Article  MathSciNet  Google Scholar 

  32. Wen, Z., Yin, W., Zhang, Y.: Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm. Math. Program. Comput. 4(4), 333–361 (2012)

    Article  MathSciNet  Google Scholar 

  33. Nesterov, Y.: Smooth minimization of non-smooth function. Math. Program. 20(6), 127–152 (2005)

    Article  MathSciNet  Google Scholar 

  34. Cai, J., Candès, F.J., Shen, Z.: A singular value thresholding decomposition. UCLA CAM Report (2010)

  35. Gandy, S., Recht, B., Yamada, I.: Tensor completion and low-n-rank tensor recovery via convex optimization. Inverse Probl. 27(2), 025010 (2011)

    Article  MathSciNet  Google Scholar 

  36. Tseng, P., Mangasarian, O.L.: Convergence of a block coordinate descent method for nondifferential minimization. J. Optim. Theory Appl. 109(3), 475–494 (2011)

    Article  Google Scholar 

  37. Xu, Z., Chang, X., Xu, F., Zhang, H.: $ L_{1/2} $ regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neur. Netw. Learn. Syst. 23(7), 1013–1027 (2012)

    Article  Google Scholar 

  38. Łojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. Les Équations aux Dérivées Partielles 117, 87–89 (1963)

    MATH  Google Scholar 

  39. Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48(3), 769–783 (1998)

    Article  MathSciNet  Google Scholar 

  40. Bertsekas, D.P.: Nonlinear Programming, 2nd edn, pp. 208–211. Athena Scientific, Nashua (1999)

    Google Scholar 

  41. Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka–Łojasiewicz inequality. Math. Oper. Res. 35(2), 438–457 (2010)

    Article  MathSciNet  Google Scholar 

  42. Yunus, A., Coskun, K., Akbar, A., Ismail, K., Fatemeh, K.: Uniform-geometric distribution. J. Stat. Comput. Simul. 86(9), 1754–1770 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, People’s Republic of China

    Shangqi Gao & Qibin Fan

Authors
  1. Shangqi Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qibin Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qibin Fan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was supported by the National Science Foundation of China under Grant 61179039.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, S., Fan, Q. Robust Schatten-p Norm Based Approach for Tensor Completion. J Sci Comput 82, 11 (2020). https://doi.org/10.1007/s10915-019-01108-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-019-01108-9

Keywords

Search

Navigation