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An Accelerated Proximal Gradient Algorithm for Hankel Tensor Completion

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Abstract

In this paper, an accelerated proximal gradient algorithm is proposed for Hankel tensor completion problems. In our method, the iterative completion tensors generated by the new algorithm keep Hankel structure based on projection on the Hankel tensor set. Moreover, due to the special properties of Hankel structure, using the fast singular value thresholding operator of the mode-s unfolding of a Hankel tensor can decrease the computational cost. Meanwhile, the convergence of the new algorithm is discussed under some reasonable conditions. Finally, the numerical experiments show the effectiveness of the proposed algorithm.

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References

  1. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: SIGGRAPH Conference (2000)

  2. Morup, M.: Applications of tensor (multiway array) factorizations and decompositions in data mining. Wiley Interdiscip. Rev. Data Min. Knowl. Discov. 1(1), 24–40 (2011)

    Article  Google Scholar 

  3. Comon, P.: Tensor decompositions, state of the art and applications. Stats. 1–24 (2009)

  4. Signoretto, M., Van de Plas, R., De Moor, B., Suykens, J.A.K.: Tensor versus matrix completion: a comparison with application to spectral data. IEEE Signal Process. Lett. 18(7), 403–406 (2011)

    Article  Google Scholar 

  5. Kolda, T.G., Bader, B.W., Kenny, J.P.: Higher-order web link analysis using multilinear algebra. In: Fifth IEEE International Conference on Data Mining, pp. 27–30 (2005)

  6. Liu, J., Musialski, P., Wonka, P., Ye, J.: Tensor completion for estimating missing values in visual data. IEEE Trans. Pattern Anal. Mach. Intell. 35(1), 208–220 (2013)

    Article  Google Scholar 

  7. Liu, J., Musialski, P., Wonka, P., Ye,J.: Tensor completion for estimating missing values in visual data. In: IEEE International Conference on Computer Vision, pp. 2114–2121 (2009)

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

    Article  MathSciNet  Google Scholar 

  9. Tan, H., Feng, J., Li, F., Zhang, Y., Chen, T.: Low multilinear rank tensor completion with missing data. Energy Procedia 11, 201–209 (2011)

    Article  Google Scholar 

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

  11. Zhang, Z., Aeron, S.: Exact tensor completion using t-SVD. IEEE Trans. Signal Process. 65(6), 1511–1526 (2017)

    Article  MathSciNet  Google Scholar 

  12. Zhang, Z., Ely, G., Aeron, S., Hao, N., Kilmer, M.: Novel methods for multilinear data completion and de-noising based on tensor-svd. In: 2014 IEEE Conference on Computer Vision and Pattern Recognition, pp. 3842–3949 (2014)

  13. Badeau, R., Boyer, R.: Fast multilinear singular value decomposition for structured tensors. Siam J. Matrix Anal. Appl. 30(3), 1008–1021 (2008)

    Article  MathSciNet  Google Scholar 

  14. Smith, R.S.: Frequency domain subspace identification using nuclear norm minimization and Hankel matrix realizations. IEEE Trans. Autom. Control 59(11), 2886–2896 (2014)

    Article  MathSciNet  Google Scholar 

  15. Vanhecke, S.V., Decannierep, C., Vanhuffel, S., Chen, H., Decabbiere, C.: Algorithm for time-domain NMR data fitting based on total least squares. J. Mag. Reson. Ser. A. 110(2), 228–237 (1994)

    Article  Google Scholar 

  16. Ding, W., Qi, L., Wei, Y.: Fast Hankel tensor–vector product and its application to exponential data fitting. Numer. Linear Algebra Appl. 22(5), 814–832 (2015)

    Article  MathSciNet  Google Scholar 

  17. Qi, L.: Hankel tensors: associated Hankel matrices and Vandermonde decomposition. Communications in Mathematical Sciences (2015). https://doi.org/10.4310/CMS.2015.v13.n1.a6

  18. Xu, C.: Hankel tensors, Vandermonde tensors and their positivities. Linear Algebra Appl. 491, 56–72 (2016)

    Article  MathSciNet  Google Scholar 

  19. Chen, Y., Qi, L., Wang, Q.: Positive semi-definiteness and sum-of-squares property of fourth order four dimensional Hankel tensors. J. Comput. Appl. Math. 302, 356–368 (2016)

    Article  MathSciNet  Google Scholar 

  20. Ding, W., Qi, L., Wei, Y.: Inheritance properties and sum-of-squares decomposition of Hankel tensors: theory and algorithms. BIT Numer. Math. 57(1), 1–22 (2016)

    MathSciNet  Google Scholar 

  21. Adamo, A., Mazzucchelli, P.: 3D interpolation using Hankel tensor completion by orthogonal mathching pursuit. In: GNGTS (2014)

  22. Trickett, S., Burroughs, L., Milton, A.: Interpolation using Hankel tensor completion. In: Seg Technical Program Expanded Abstracts, pp. 3634–3638 (2013)

  23. Toh, C.K., Yun, 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 

  24. Comon, P., Golub, G., Lim, L.H., Mourrain, B.: Symmetric tensors and symmetric tensor rank. Siam J. Matrix Anal. Appl. 30(3), 1254–1279 (2008)

    Article  MathSciNet  Google Scholar 

  25. Nie, J., Ye, K.: Hankel tensor decompositions and ranks. SIAM J. Matrix Anal. Appl. 40(2), 486–516 (2019)

    Article  MathSciNet  Google Scholar 

  26. Golub, G.H., VanLoan, C.F.: Matrix Computations. The Johns Hopkins University Press, Baltimore (1996)

    Google Scholar 

  27. Cai, J.F., Candès, E.J., Shenm, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)

    Article  MathSciNet  Google Scholar 

  28. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imag. Sci. 2(1), 183–202 (2009)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of Engineering and Computational Science, Shanxi Provincial Department of Education, Jinzhong, 030619, Shanxi, China

    Chuan-Long Wang & Xiong-Wei Guo

  2. Department of Mathematics, Taiyuan Normal University, Jinzhong, 030619, Shanxi, China

    Chuan-Long Wang, Xiong-Wei Guo & Xi-Hong Yan

Authors
  1. Chuan-Long Wang
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  2. Xiong-Wei Guo
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  3. Xi-Hong Yan
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Contributions

Chuan-long Wang: Conceptualization, Methodology, Supervision; Xiong-wei Guo: Software, Writing - Original Draft, Writing - Review & Editing; Xi-hong Yan: Funding acquisition, Project administration, Supervision.

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Correspondence to Xi-Hong Yan.

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Wang, CL., Guo, XW. & Yan, XH. An Accelerated Proximal Gradient Algorithm for Hankel Tensor Completion. J. Oper. Res. Soc. China (2022). https://doi.org/10.1007/s40305-022-00422-8

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  • DOI: https://doi.org/10.1007/s40305-022-00422-8

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