Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Calcolo
Published in: Calcolo 2/2024

01-06-2024

Tensor completion via multi-directional partial tensor nuclear norm with total variation regularization

Authors: Rong Li, Bing Zheng

Published in: Calcolo | Issue 2/2024

Login to get access

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This paper addresses the tensor completion problem, whose task is to estimate missing values with limited information. However, the crux of this problem is how to reasonably represent the low-rank structure embedded in the underlying data. In this work, we consider a new low-rank tensor completion model combined with the multi-directional partial tensor nuclear norm and the total variation (TV) regularization. Specifically, the partial sum of the tensor nuclear norm (PSTNN) is used to narrow the gap between the tensor tubal rank and its lower convex envelop [i.e. tensor nuclear norm (TNN)], and the TV regularization is adopted to maintain the smooth structure along the spatial dimension. In addition, the weighted sum of the tensor nuclear norm (WSTNN) is introduced to replace the traditional TNN to extend the PSTNN to the high-order tensor, which also can flexibly handle different correlations along different modes, resulting in an improved low d-tubal rank approximation. To tackle this new model, we develop the alternating directional method of multipliers (ADMM) algorithm tailored for the proposed optimization problem. Theoretical analysis of the ADMM is conducted to prove the Karush–Kuhn–Tucker (KKT) conditions. Numerical examples demonstrate the proposed method outperforms some state-of-the-art methods in qualitative and quantitative aspects.
next article An efficient and unified method for band structure calculations of 2D anisotropic photonic-crystal fibers
Footnotes
Literature
1.
Bagirov, A., Karmitsa, N., Marko, M.M.: Nonconvex Analysis. Introduction to Nonsmooth Optimization. Springer, London (2014)CrossRef
2.
Cai, J.-F., Candès, E.J., Shen, Z.-W.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)MathSciNetCrossRef
3.
Carroll, J.D., Pruzansky, S., Kruskal, J.B.: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters. Psychometrika 45(1), 3–24 (1980)MathSciNetCrossRef
4.
Chen, C.-H., He, B.-S., Yuan, X.-M.: Matrix completion via an alternating direction method. IMA J. Numer. Anal. 32(1), 227–245 (2012)MathSciNetCrossRef
5.
Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)CrossRef
6.
Comon, P.: Tensor decompositions: state of the art and applications. Math. Signal Proces, Inst. Math. Appl. Conf. Ser. New Ser. 71. Oxford University Press, Oxford (2001)
7.
Fazel, M.: Matrix rank minimization with applications. Ph.D. Thesis, Stanford University, Stanford, CA (2002)
8.
Gandy, S., Recht, B., Yamada, I.: Tensor completion and low-\(n\)-rank tensor recovery via convex optimization. Inverse Prob. 27(2), 025010 (2011)ADSMathSciNetCrossRef
9.
Gao, S.-Q., Zhuang, X.-H.: Robust approximations of low-rank minimization for tensor completion. Neurocomputing 379, 319–333 (2020)CrossRef
10.
11.
Ji, T.-Y., Huang, T.-Z., Zhao, X.-L., Ma, T.-H., Deng, L.-J.: A non-convex tensor rank approximation for tensor completion. Appl. Math. Model. 48, 410–422 (2017)MathSciNetCrossRef
12.
Ji, T.-Y., Huang, T.-Z., Zhao, X.-L., Ma, T.-H., Liu, G.: Tensor completion using total variation and low-rank matrix factorization. Inf. Sci. 326, 243–257 (2016)MathSciNetCrossRef
13.
Jiang, T.-X., Huang, T.-Z., Zhao, X.-L., Deng, L.-J.: Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm. J. Comput. Appl. Math. 372, 112680 (2020)MathSciNetCrossRef
14.
Jiang, T.-X., Huang, T.-Z., Zhao, X.-L., Deng, L.-J., Wang, Y.: Fastderain: a novel video rain streak removal method using directional gradient priors. IEEE Trans. Image Process. 28(4), 2089–2102 (2018)ADSMathSciNetCrossRef
15.
Kajo, I., Kamel, N., Ruichek, Y., Malik, A.S.: SVD-based tensor-completion technique for background initialization. IEEE Trans. Image Process. 27(6), 3114–3126 (2018)ADSMathSciNetCrossRefPubMed
16.
Kang, Z., Peng, C., Cheng, Q.: Robust PCA via nonconvex rank approximation. In: IEEE International Conference on Data Mining, pp. 211–220 (2015)
17.
Karatzoglou, A., Amatriain, X., Baltrunas, L., Oliver, N.: Multiverse recommendation: \(n\)-dimensional tensor factorization for context-aware collaborative filtering. In: Proceedings of the 4th ACM Conference Recommender Systems. ACM, New York (2010)
18.
Kilmer, M.E., Braman, K., Hao, N., Hoover, R.C.: Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging. SIAM J. Matrix Anal. Appl. 34(1), 148–172 (2013)MathSciNetCrossRef
19.
Kilmer, M.E., Martin, C.D.: Factorization strategies for third-order tensor. Linear Algebra Appl. 435(3), 641–658 (2011)MathSciNetCrossRef
20.
21.
Kreimer, N., Sacchi, M.D.: A tensor higher-order singular value decomposition for prestack seismic data noise reduction and interpolation. Geophysics 77(3), 113–122 (2012)ADSCrossRef
22.
Li, X.-T., Ye, Y.-M., Xu, X.-F.: Low-rank tensor completion with total variation for visual data in painting. AAA I, 2210–2216 (2017)
23.
Lieven, D.L., Bart, D.M.: From matrix to tensor: multilinear algebra and signal processing. Math. Signal Process. Inst. Math. Appl. Conf. Ser. New Ser. 71. Oxford University Press, Oxford (1996)
24.
Lin, X.-L., Ng, M.K., Zhao, X.-L.: Tensor factorization with total variation and Tikhonov regularization for low-rank tensor completion in imaging data. J. Math. Imaging Vis. 62, 900–918 (2020)MathSciNetCrossRef
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), 208–220 (2013)CrossRefPubMed
26.
Lu, C.-Y., Feng, J.-S., Chen, Y.-D., Liu, W., Lin, Z.-C., Yan, S.-C.: Tensor robust principal component analysis with a new tensor nuclear norm. IEEE Trans. Pattern Anal. Mach. Intell. 42(4), 925–938 (2020)CrossRefPubMed
27.
Mian, A., Hartley, R.: Hyperspectral video restoration using optical flow and sparse coding. Opt. Express 20(10), 10658–10673 (2012)ADSCrossRefPubMed
28.
29.
Mørup, M.: Applications of tensor (multiway array) factorizations and decompositions in data mining. Wiley Interdiscip. Rev. Data Min. Knowl. Discov. 1(1), 24–40 (2011)CrossRef
30.
Mu, C., Huang, B., Wright, J., Goldfarb, D.: Square deal: lower bounds and improved relaxations for tensor recovery. In: In the International Conference on Machine Learning (ICML), pp. 73–81 (2014)
31.
Ng, M.K., Yuan, Q.-Q., Yan, L., Sun, J.: An adaptive weighted tensor completion method for the recovery of remote sensing images with missing data. IEEE Trans. Geosci. Remote Sens. 55(6), 3367–3381 (2017)ADSCrossRef
32.
Pan, P., Wang, Y.-L., Chen, Y.-Y., Wang, S.-Q., He, G.-P.: A new nonconvex rank approximation of RPCA. Sci. Tech. Eng. 17(31), 1671–1815 (2017)
33.
34.
Sauve, A.C., Hero, A.O., Rogers, W.L., Wilderman, S.J., Clinthorne, N.H.: 3D image reconstruction for a Compton Spect Camera model. IEEE Trans. Nucl. Sci. 46(6), 2075–2084 (1999)ADSCrossRef
35.
Silva, V.D., Lim, L.H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084–1127 (2008)MathSciNetCrossRef
36.
Sun, J.-T., Zeng, H.-J., Liu, H., Lu, Y.-C., Zheng, C.: CubeSVD: a novel approach to personalized web search. In: Proc. 14th Int. Conf. World Wide Web, pp. 382–390 (2005)
37.
Varghees, V.N., Manikandan, M.S., Gini, R.: Adaptive MRI image denoising using total-variation and local noise estimation. In: Proceedings of the 2012 International Conference on Advances in Engineering, Science and Management (ICAESM), pp. 506–511 (2012)
38.
Wang, B., Gao, X.-B., Tao, D.-C., Li, X.-L.: A unified tensor level set for image segmentation. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 40(3), 857–867 (2010)CrossRef
39.
Xie, Q., Zhao, Q., Meng, D.-Y., Xu, Z.-B.: Kronecker-basis-representation based tensor sparsity and its applications to tensor recovery. IEEE Trans. Pattern Anal. Mach. Intell. 40(8), 1888–1902 (2018)CrossRefPubMed
40.
Xue, J.-Z., Zhao, Y.-Q., Liao, W.-Z., Chan, C.-W.: Nonconvex tensor rank minimization and its applications to tensor recovery. Inf. Sci. 503, 109–128 (2019)MathSciNetCrossRef
41.
Yang, J.-H., Zhao, X.-L., Ma, T.-H., Chen, Y., Huang, T.-Z., Ding, M.: Remote sensing images destriping using unidirectional hybrid total variation and nonconvex low-rank regularization. J. Comput. Appl. Math. 363(4), 124–144 (2020)MathSciNetCrossRef
42.
Zhang, L., Zhang, L., Mou, X.-Q., Zhang, D.: FSIM: a feature similarity index for image quality assessment. IEEE Trans. Image Process. 20(8), 2378–2386 (2011)ADSMathSciNetCrossRefPubMed
43.
44.
Zhao, Q.-B., Zhang, L.-Q., Cichocki, A.: Bayesian CP factorization of incomplete tensors with automatic rank determination. IEEE Trans. Pattern Anal. Mach. Intell. 37(9), 1751–1763 (2015)CrossRefPubMed
45.
Zhao, X.-Y., Bai, M.-R., Ng, M.K.: Nonconvex optimization for robust tensor completion from grossly sparse observations. J. Sci. Comput. 85(2), 1–32 (2020)MathSciNetCrossRef
46.
Zheng, Y.-B., Huang, T.-Z., Zhao, X.-L., Jiang, T.-X., Ji, T.-Y., Ma, T.-H.: Tensor N-tubal rank and its convex relaxation for low-rank tensor recovery. Inf. Sci. 532, 170–189 (2020)MathSciNetCrossRef
47.
Zhou, M.-Y., Liu, Y.-P., Long, Z., Chen, L.-X., Zhu, C.: Tensor rank learning in CP decomposition via convolutional neural network. Signal Process Image Commun. 73(3), 12–21 (2019)CrossRef
48.
Zhou, W., Bovik, A., Sheikh, H., Simoncelli, E.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)ADSCrossRef
Metadata
Title
Tensor completion via multi-directional partial tensor nuclear norm with total variation regularization
Authors
Rong Li
Bing Zheng
Publication date
01-06-2024
Publisher
Springer International Publishing
Published in
Calcolo / Issue 2/2024
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-024-00569-1

Other articles of this Issue 2/2024

Calcolo 2/2024 Go to the issue

OriginalPaper

An asymptotic preserving scheme for the model on polygonal and conical meshes

OriginalPaper

Discrete projection methods for Fredholm–Hammerstein integral equations using Kumar and Sloan technique

OriginalPaper

Greedy randomized sampling nonlinear Kaczmarz methods

OriginalPaper

Weight calculation and convergence analysis of polyharmonic spline (PHS) with polynomials for different stencils

OriginalPaper

An efficient and unified method for band structure calculations of 2D anisotropic photonic-crystal fibers

OriginalPaper

Numerical approximation of the stochastic Navier–Stokes equations through artificial compressibility

Premium Partner

    Image Credits