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Fast multidimensional completion and principal component analysis methods via the cosine product

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Abstract

In the present work, we propose new methods for the problem of third-order tensor completion and tensor robust principal component analysis (TRPCA). The proposed approaches are based on finding a low-rank tensor by solving some optimization problems of tensor nuclear norm under some constraints and by using the discrete cosine transform. For the problem of completion, we add some regularization techniques by using the first order and a second-order total variation to enhance the results. Both the main optimization problems lead to some tensor problems that will be solved by using the Alternative Direction Method of Multipliers (ADMM), and also we use for the problem of TRPCA the Proximal Gradient Algorithm to solve it and we will compare the results given by the two ways. We also present some numerical experiments of the proposed methods.

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References

  1. Aeron, S., Ely, G., Hoa, N., Kilmer, M., Zhang, Z.: Novel methods for multilinear data completion and de-noising based on tensor-SVD. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 3842–3849 (2014)

  2. Aeron, S., Kernfeld, E., Kilmer, M.: Tensor-tensor products with invertible linear transforms. Linear Algebra Appl. 485, 545–570 (2015)

    Article  MathSciNet  Google Scholar 

  3. Bader, B.W., Joseph, J.P., Kolda, T.G.: Higher-order web link analysis using multilinear algebra. In: Fifth IEEE International Conference on Data Mining (ICDM’05), 8 pp (2005)

  4. Ballester, C., Bertalmio, M., Caselles, V., Sapiro, G.: Image inpainting. In: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pp. 417–424 (2000)

  5. Barder, B.W., Kolda, T.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009)

    Article  MathSciNet  Google Scholar 

  6. Beck, A.: First-order methods in optimization. MOS-SIAM Series on Optimization. SIAM (2017). https://doi.org/10.1137/1.9781611974997

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

    Article  MathSciNet  Google Scholar 

  8. Benczúr, A.A., Csalogány, K., Kurucs, M.: Methods for large scale SVD with missing values. Proc. KDD Cup Workshop 12, 31–38 (2007)

    Google Scholar 

  9. Bentbib, A.H., El Guide, M., Jbilou, K.: A generalized matrix Krylov subspace method for TV regularization. J. Comput. Appl. Math. 373, 112405 (2020)

    Article  MathSciNet  Google Scholar 

  10. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 3(1):1–122 (2011)

    Article  Google Scholar 

  11. Boyd, S.P., Fazel, M., Hindi, H.: A rank minimization heuristic with application to minimum order system approximation. In: Proceedings of the 2001 American Control Conference. (Cat. No. 01CH37148), vol. 6, pp. 4734–4739 (2001)

  12. Braman, K., Hao, N., Hoover, R.C., Kilmer, M.E.: Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging. SIAM J. Matrix Anal. Appl. 34, 148–172 (2013)

    Article  MathSciNet  Google Scholar 

  13. Calatroni, L., Lanza, A., Pragliola, M., Sgallari, F.: Adaptive parameter selection for weighted-TV image reconstruction problems. J. Phys.: Conf. Ser. 1476, 012003 (2020)

    Google Scholar 

  14. Calatroni, L., Lanza, A., Pragliola, M., Sgallari, F.: A flexible space-variant anisotropic regularization for image restoration with automated parameter selection. SIAM J. Imaging Sci. 12, 1001–1037 (2019)

    Article  MathSciNet  Google Scholar 

  15. Candes, E.J., Rechet, B.: Exact low-rank matrix completion via convex optimization. In: 2008 46th Annual Allerton Conference on Communication, Control, and Computing, pp. 806–812 (2008)

  16. Chen, Y., Huang, T.-Z., Zhao, X.-L.: Destriping of multispectral remote sensing image using low-rank tensor decomposition. IEEE J. Select. Topics Appl. Earth Observ. Remote Sens. 11, 4950–4967 (2018)

    Article  Google Scholar 

  17. Chen, Y., Feng, J., Lin, H., Liu, W., Lu, C., Yan, S.: Tensor robust principal component analysis with a new tensor nuclear norm. IEEE Trans. Pattern Anal. Mach. Intell. 42, 925–938 (2019)

    Google Scholar 

  18. Deng, L.-J., Huang, T.-Z., Ji, T.-Y., Jiang, T.-X., Zhao, X.-L.: Matrix factorization for low-rank tensor completion using framelet prior. Inf. Sci. 436, 403–417 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Ding, M., Huang, T.-Z., Ji, T.-Y., Yang, J.-H., Zhao, X.-L.: Low-rank tensor completion using matrix factorization based on tensor train rank and total variation. J. Sci. Comput. 81, 941–964 (2019)

    Article  MathSciNet  Google Scholar 

  20. Dong, W., Fu, Y.: 3D magnetic resonance image denoising using low-rank tensor approximation. Neurocomputing 195, 30–39 (2016)

    Article  Google Scholar 

  21. El Guide, M., El Ichi, A., Jbilou, K., Sadaka, R.: Tensor Krylov subspace methods via the T-product for color image processing. Electron. Linear Algebra 37, 524–543 (2021)

    Article  MathSciNet  Google Scholar 

  22. El Guide, M., El Ichi, A., Jbilou, K.: Discrete cosine transform LSQR methods for multidimensional ill-posed problems. J. Math. Model. 10(1), 21–37 (2021)

    MathSciNet  MATH  Google Scholar 

  23. Facchinei, F., Pang, J.-S.: Finite-dimensional variational inequalities and complementarity problems. Springer, New York (2003)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Goldfarb, D., Qin, Z.: Robust low-rank tensor recovery: models and algorithms. SIAM J. Matrix Anal. Appl. 35, 225–253 (2014)

    Article  MathSciNet  Google Scholar 

  26. Hillar, C.J., Lim, L.-H.: Most tensor problems are NP-hard. J. ACM (JACM) 60, 1–39 (2013)

    Article  MathSciNet  Google Scholar 

  27. Huang, T.-Z., Ji, T.-Y., Liu, G., Ma, T.-H., Zhao, X.-L.: Tensor completion using total variation and low-rank matrix factorization. Inf. Sci. 326, 243–257 (2016)

    Article  MathSciNet  Google Scholar 

  28. Ji, T.-Y., Jiang, T.-X., Huang, T.-Z., Ma, T.-H., Zhao, X.-L., Zheng, Y.-B.: Tensor N-tubal rank and its convex relaxation for low-rank tensor recovery. Inf. Sci. 532, 170–189 (2020)

    Article  MathSciNet  Google Scholar 

  29. Komodakis, N.: Image completion using global optimization. In: 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), vol. 1, pp. 442–452 (2006)

  30. Li, X., Huang, T.-Z., Zhao, X.-L., Ji, T.-Y., Zheng, Y.-B., Deng, L.-J.: Adaptive total variation and second-order total variation-based model for low-rank tensor completion. Numer. Algorithms 86, 1–24 (2021)

    Article  MathSciNet  Google Scholar 

  31. Li, F., Ng, M.K., Robert, R.J.: Coupled segmentation and denoising/deblurring models for hyperspectral material identification. Numer. Linear Algebra Appl. 19, 153–173 (2012)

    Article  MathSciNet  Google Scholar 

  32. 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. In: Coordinated Science Laboratory Report no. UILU-ENG-09-2214, DC-246 (2009)

  33. Ng, M., Xu, W.-H., Zhao, X.-L.: A fast algorithm for cosine transform based tensor singular value decomposition (2019). arXiv preprint arXiv:1902.03070

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

    Article  MathSciNet  Google Scholar 

  35. Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton (2015)

    Google Scholar 

  36. Rolant, R., Manikandan, M.S., Varghees, V.N.: Adaptive MRI image denoising using total-variation and local noise estimation. In: IEEE-International Conference On Advances In Engineering. Science And Management (ICAESM-2012), pp. 506–511 (2012)

  37. Tai, X.-C., Wu, C., Zhang, J.: Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems & Imaging. Am. Inst. Math. Sci. 5, 237 (2011)

    MATH  Google Scholar 

  38. Tseng, P.: On accelerated proximal gradient methods for convex-concave optimization. SIAM J. Optimiz. 2 (2008)

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Authors and Affiliations

  1. Faculté des Sciences et Techniques-Gueliz, Laboratoire de Mathématiques Appliquées, Marrakech, Morocco

    A. H. Bentbib

  2. Laboratory MSDA, Mohammed VI Polytechnic University, Green City, Morocco

    A. El Hachimi, K. Jbilou & A. Ratnani

  3. Université du Littoral Cote d’Opale, LMPA, 50 rue F. Buisson, 62228, Calais-Cedex, France

    K. Jbilou

  4. University M6, Benguérir, Morocco

    K. Jbilou

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  1. A. H. Bentbib
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  2. A. El Hachimi
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  3. K. Jbilou
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  4. A. Ratnani
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Correspondence to K. Jbilou.

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Bentbib, A.H., El Hachimi, A., Jbilou, K. et al. Fast multidimensional completion and principal component analysis methods via the cosine product. Calcolo 59, 26 (2022). https://doi.org/10.1007/s10092-022-00469-2

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  • DOI: https://doi.org/10.1007/s10092-022-00469-2

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