Skip to main content
SpringerLink
Log in

Robust matrix completion with complex noise

  • Published:
Multimedia Tools and Applications Aims and scope Submit manuscript

Abstract

Matrix completion plays an important role in machine learning and data mining. Although a great number of algorithms have been developed for this issue, most of them can cope with only the Gaussian noise or sparse outliers. This paper focus on an intractable setting that the known entries are corrupted by Gaussian noise and sparse outliers simultaneously. Specifically, we construct a novel model with a loss function derived from the celebrated Huber function. Furthermore, an efficient optimization method is presented to solve the constructed model. The promising performance of our algorithm is demonstrated via numerous experiments on several benchmark datasets.

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

Similar content being viewed by others

References

  1. Agudo A, Pijoan M, Moreno-Noguer F (2018) Image collection pop-up: 3d reconstruction and clustering of rigid and non-rigid categories. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2607–2615

  2. Basri R, Jacobs D, Kemelmacher I (2007) Photometric stereo with general, unknown lighting. Int J Comput Vis 72(3):239–257

    Article  Google Scholar 

  3. Cai J-F, Candès EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982

    Article  MathSciNet  Google Scholar 

  4. Cambier L, Absil P-A (2016) Robust low-rank matrix completion by riemannian optimization. SIAM J Sci Comput 38(5):S440–S460

    Article  MathSciNet  Google Scholar 

  5. Candès EJ, Li X, Ma Y, Wright J (2011) Robust principal component analysis? J ACM (JACM) 58(3):11

    Article  MathSciNet  Google Scholar 

  6. Candès EJ, Tao T (2010) The power of convex relaxation: near-optimal matrix completion. IEEE Trans Inf Theory 56(5):2053–2080

    Article  MathSciNet  Google Scholar 

  7. Dai Y, Li H, He M (2014) A simple prior-free method for non-rigid structure-from-motion factorization. Int J Comput Vis 107(2):101–122

    Article  MathSciNet  Google Scholar 

  8. Elsener A, Van De Geer S (2016) Robust low-rank matrix estimation

  9. Eriksson A, Van Den Hengel A (2010) Efficient computation of robust low-rank matrix approximations in the presence of missing data using the l1 norm

  10. Goldberg K, Roeder T, Gupta D, Perkins C (2001) Eigentaste: a constant time collaborative filtering algorithm. Information Retrieval 4(2):133–151

    Article  Google Scholar 

  11. Gu S, et al. (2017) Weighted nuclear norm minimization and its applications to low level vision. Int J Comput Vis 121(2):183–208

    Article  Google Scholar 

  12. He J, Balzano L, Szlam A (2012) Incremental gradient on the grassmannian for online foreground and background separation in subsampled video. In: 2012 IEEE conference on computer vision and pattern recognition (CVPR). IEEE, pp 1568–1575

  13. Herlocker JL, Konstan JA, Borchers A, Riedl J (2017) An algorithmic framework for performing collaborative filtering. In: ACM SIGIR forum, vol 51. ACM, pp 227–234

  14. Hu Y, Zhang D, Ye J, Li X, He X (2013) Fast and accurate matrix completion via truncated nuclear norm regularization. IEEE Trans Pattern Anal Mach Intell 35(9):2117–2130

    Article  Google Scholar 

  15. Hu Z, Nie F, Wang R, Li X (2019) Multi-view spectral clustering via integrating nonnegative embedding and spectral embedding. Inf Fusion

  16. Huber PJ (2011) Robust statistics. In: International encyclopedia of statistical science. Springer, pp 1248–1251

  17. Jegou H, Douze M, Schmid C (2008) Hamming embedding and weak geometric consistency for large scale image search. In: European conference on computer vision. Springer, pp 304–317

  18. Ke Q, Kanade T (2005) Robust ℓ1 norm factorization in the presence of outliers and missing data by alternative convex programming. In: IEEE computer society conference on computer vision and pattern recognition, 2005. CVPR 2005, vol 1. IEEE, pp 739–746

  19. Lin Z, Xu C, Zha H (2018) Robust matrix factorization by majorization minimization. IEEE Trans Pattern Anal Mach Intell 40(1):208–220

    Article  Google Scholar 

  20. Liu G, et al. (2013) Robust recovery of subspace structures by low-rank representation. IEEE Trans Pattern Anal Mach Intell 35(1):171–184

    Article  Google Scholar 

  21. Liu Q, Davoine F, Yang J, Cui Y, Jin Z, Han F (2019) A fast and accurate matrix completion method based on QR decomposition and ℓ2,1 -norm minimization. IEEE Trans Neural Netw Learning Syst 30(3):803–817

    Article  MathSciNet  Google Scholar 

  22. Lu C, Tang J, Yan S, Lin Z (2014) Generalized nonconvex nonsmooth low-rank minimization. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 4130–4137

  23. Lu C, Zhu C, et al. (2015) Generalized singular value thresholding. In: AAAI, pp 1805–1811

  24. Meng D, Xu Z, Zhang L, Ji Z (2013) A cyclic weighted median method for l1 low-rank matrix factorization with missing entries. In: AAAI, vol 4, p 6

  25. Ngo T, Saad Y (2012) Scaled gradients on grassmann manifolds for matrix completion. In: Advances in neural information processing systems, pp 1412–1420

  26. Nie F, Hu Z, Li X (2018) Calibrated multi-task learning. In: Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining . ACM, pp 2012–2021

  27. Nie F, Hu Z, Li X (2019) Matrix completion based on non-convex low-rank approximation. IEEE Trans Image Process 28(5):2378–2388

    Article  MathSciNet  Google Scholar 

  28. Nie F, Huang H, Ding CHQ (2012) Low-rank matrix recovery via efficient schatten p-norm minimization. In: AAAI

  29. Philbin J, Chum O, Isard M, Sivic J, Zisserman A (2007) Object retrieval with large vocabularies and fast spatial matching. In: IEEE conference on computer vision and pattern recognition, 2007. CVPR’07. IEEE, p 2007

  30. Pierre A, Vincent C, Guillaume L (2019) Estimation bounds and sharp oracle inequalities of regularized procedures with lipschitz loss functions. The Annals of Statistics 47(4):2117–2144

    Article  MathSciNet  Google Scholar 

  31. Kwok J, Yao QM (2018) Scalable robust matrix factorization with nonconvex loss. In: Proceedings of the advances in neural information processing systems, pp 369–404

  32. Shang F, et al. (2017) Bilinear factor matrix norm minimization for robust pca: algorithms and applications. IEEE Trans Pattern Anal Mach Intell (1):1–1

  33. Shang F, Liu Y, Cheng J, Yan D (2018) Fuzzy double trace norm minimization for recommendation systems. IEEE Trans Fuzzy Systems 26(4):2039–2049

    Article  Google Scholar 

  34. Shang F, Liu Y, Tong H, Cheng J, Cheng H (2015) Robust bilinear factorization with missing and grossly corrupted observations. Inf Sci 307:53–72

    Article  MathSciNet  Google Scholar 

  35. Sun Q, Xiang S, Ye J (2013) Robust principal component analysis via capped norms. In: Proceedings of the 19th ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 311–319

  36. Tanner J, Ke W (2016) Low rank matrix completion by alternating steepest descent methods. Appl Comput Harmon Anal 40(2):417–429

    Article  MathSciNet  Google Scholar 

  37. Tomasi C, Kanade T (1992) Shape and motion from image streams under orthography: a factorization method. Int J Comput Vis 9(2):137–154

    Article  Google Scholar 

  38. Vandereycken B (2013) Low-rank matrix completion by riemannian optimization. SIAM J Optim 23(2):1214–1236

    Article  MathSciNet  Google Scholar 

  39. Wang Y, Wu L, Lin X, Gao J (2018) Multiview spectral clustering via structured low-rank matrix factorization. IEEE Trans Neural Netw Learning Syst 29 (10):4833–4843

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  41. Yao Q (2018) Scalable tensor completion with nonconvex regularization. arXiv:1807.08725

  42. Yao Q, Kwok JT, Wang T, Liu T-Y (2017) Large-scale low-rank matrix learning with nonconvex regularizers. arXiv:1708.00146

  43. Zhao L, Babu P, Palomar DP (2016) Efficient algorithms on robust low-rank matrix completion against outliers. IEEE Trans Signal Processing 64(18):4767–4780

    Article  MathSciNet  Google Scholar 

  44. Zheng Y, Liu G, Sugimoto S, Yan S, Okutomi M (2012) Practical low-rank matrix approximation under robust l 1-norm. In: 2012 IEEE conference on computer vision and pattern recognition (CVPR). IEEE, p 2012

  45. Zhu L, Zi H, Li Z, Xie L, Shen HT (2018) Exploring auxiliary context: discrete semantic transfer hashing for scalable image retrieval. IEEE Trans Neural Netw Learning Syst 29(11):5264–5276

    Article  MathSciNet  Google Scholar 

  46. Zhu L, Zi H, Liu X, He X, Sun J, Zhou X (2017) Discrete multimodal hashing with canonical views for robust mobile landmark search. IEEE Trans Multimedia 19(9):2066–2079

    Article  Google Scholar 

  47. Zhu L, Shen J, Xie L, Cheng Z (2017) Unsupervised visual hashing with semantic assistant for content-based image retrieval. IEEE Trans Knowl Data Eng 29 (2):472–486

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Teacher Education Institute, Daqing Normal University, Xibin Xi Road, Ranghulu District, Daqing, 163712, Heilongjiang, China

    Li Tang

  2. Hewlett Packard Enterprise Singapore, Singapore, Singapore

    Weili Guan

Authors
  1. Li Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weili Guan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Tang.

Additional information

Publisher’s note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, L., Guan, W. Robust matrix completion with complex noise. Multimed Tools Appl 79, 2703–2717 (2020). https://doi.org/10.1007/s11042-019-08430-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11042-019-08430-2

Keywords

Search

Navigation