Adaptive algorithms for low-rank and sparse matrix recovery with truncated nuclear norm

  • Wenchao Qian
  • Feilong Cao
Original Article


Recent studies have shown that the use of the truncated nuclear norm (TNN) in low-rank and sparse matrix decomposition (LRSD) can realize a better approximation to rank function of matrix, and achieve effectively recovery effects. This paper addresses the algorithms for LRSD with adaptive TNN (LRSD-ATNN), and designs an efficient algorithmic frame inspired by the alternating direction method of multiple (ADMM) and the accelerated proximal gradient approach (APG). To establish the adaptive algorithms, the method of singular value estimate is utilized to find adaptively the number of truncated singular value. Experimental results on synthetic data as well as real visual data show the superiority of the proposed algorithm in effectiveness in comparison with the state-of-the-art methods.


Low-rank and sparse matrix decomposition Truncated nuclear norm Adaptive algorithm Alternating direction method of multiple Accelerated proximal gradient 



This research was supported by the National Natural Science Foundation of China under Grant 61672477.


  1. 1.
    Bach F, Jenatton R, Mairal J, Obozinski G (2012) Optimization with sparsity-inducing penalties. Found Trends Mach Learn 4(1):1–106CrossRefzbMATHGoogle Scholar
  2. 2.
    Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci 2(1):183–202MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bouwmans T, Zahzah H E (2014) Robust PCA via principal component pursuit: A review for a comparative evaluation in video surveillance. Comput Vis Image Underst 122:22–34CrossRefGoogle Scholar
  4. 4.
    Cai JF, Candès EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Candès EJ, Li X, Ma Y, Wright J (2011) Robust principal component analysis? J ACM 58(3):1–37MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Candès EJ, Recht B (2009) Exact matrix completion via convex optimization. Found Comput Math 9(6):717–772MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Candès EJ, Romberg J, Tao T (2006) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52(2):489–509MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Candès EJ, Tao T (2005) Decoding by linear programming. IEEE Trans Inf Theory 51(12):4203–4215MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Candès EJ, Tao T (2010) The power of convex relaxation: near-optimal matrix completion. IEEE Trans Inf Theory 56(5):2053–2080MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cao F, Chen J, Ye H, Zhao J, Zhou Z (2017) Recovering low-rank and sparse matrix based on the truncated nuclear norm. Neural Netw 85:10–20CrossRefGoogle Scholar
  11. 11.
    Cao W, Wang Y, Sun J, Meng D, Yang C, Cichocki A, Xu Z (2016) Total variation regularized tensor RPCA for background subtraction from compressive measurements. IEEE Trans Image Process 25(9):4075–4090MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cao X, Yang L, Guo X (2016) Total variation regularized RPCA for irregularly moving object detection under dynamic background. IEEE Trans Cybern 46(4):1014–1027CrossRefGoogle Scholar
  13. 13.
    Dang C, Radha H (2015) RPCA-KFE: key frame extraction for video using robust principal component analysis. IEEE Trans Image Process 24(11):3742–3753MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ebadi SE, Izquierdo E (2017) Foreground segmentation with tree-structured sparse RPCA. IEEE Trans Pattern Anal Mach Intell. Google Scholar
  15. 15.
    Fazel M (2002) Matrix rank minimization with applications. Doctoral dissertation, PhD thesis, Stanford UniversityGoogle Scholar
  16. 16.
    F’evotte C, Gribonval R, Vincent E (2005) BSS_EVAL toolbox user guide—revision 2.0. Technical report 1706, IRISA, pp 19. Accessed 15 Dec 2017
  17. 17.
    Fu Z, Wang X, Xu J, Zhou N, Zhao Y (2016) Infrared and visible images fusion based on RPCA and NSCT. Infrared Phys Technol 77:114–123CrossRefGoogle Scholar
  18. 18.
    Fukushima M, Mine H (1981) A generalized proximal gradient algorithm for certain nonconvex minimization problems. Int J Syst Sci 12(8):989–1000CrossRefzbMATHGoogle Scholar
  19. 19.
    Ganesh A, Lin Z, Wright J, Wu L, Chen M, Ma Y (2009) Fast algorithms for recovering a corrupted low-rank matrix. In: Proceedings of 3rd IEEE international workshop on computational advances in multi-sensor adaptive processing (CAMSAP). IEEE, Aruba, pp 213–216Google Scholar
  20. 20.
    Gu S, Xie Q, Meng D, Zuo W, Feng X, Zhang L (2017) Weighted nuclear norm minimization and its applications to low level vision. Int J Comput Vis 121(2):183–208CrossRefGoogle Scholar
  21. 21.
    Hale ET, Yin W, Zhang Y (2008) Fixed-point continuation for minimization: methodology and convergence. SIAM J Optim 19(3):1107–1130MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Han ZF, Leung CS, Huang LT, So HC (2017) Sparse and truncated nuclear norm based tensor completion. Neural Process Lett 45(3):729–743CrossRefGoogle Scholar
  23. 23.
    He Y, Li M, Zhang J, An Q (2015) Small infrared target detection based on low-rank and sparse representation. Infrared Phys Technol 68:98–109CrossRefGoogle Scholar
  24. 24.
    Hong B, Wei L, Hu Y, Cai D, He X (2016) Online robust principal component analysis via truncated nuclear norm regularization. Neurocomputing 175:216–222CrossRefGoogle Scholar
  25. 25.
    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):2117–2130CrossRefGoogle Scholar
  26. 26.
    Huang PS, Chen SD, Smaragdis P, Hasegawa-Johnson M (2012) Singing-voice separation from monaural recordings using robust principal component analysis. In: Proceedings of 2012 IEEE international conference on acoustics, speech and signal processing (ICASSP). IEEE, Kyoto, Japan, pp 57–60Google Scholar
  27. 27.
    Javed S, Bouwmans T, Shah M, Jung SK (2017) Moving object detection on RGB-D videos using graph regularized spatiotemporal RPCA. In: Proceedings of 2017 International conference on image analysis and processing (ICIAP). Springer, Berlin, pp 230–241Google Scholar
  28. 28.
    Jin M, Li R, Jiang J, Qin B (2017) Extracting contrast-filled vessels in X-ray angiography by graduated RPCA with motion coherency constraint. Pattern Recogn 63:653–666CrossRefGoogle Scholar
  29. 29.
    Lin L, Lin W, Huang S (2018) Group object detection and tracking by combining RPCA and fractal analysis. Soft Comput 22(1):231–242CrossRefGoogle Scholar
  30. 30.
    Lin Z, Chen M, Ma Y (2010) The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. arXiv preprint arXiv:1009.5055Google Scholar
  31. 31.
    Lin Z, Ganesh A, Wright J, Wu L, Chen M, Ma Y (2009) Fast convex optimization algorithms for exact recovery of a corrupted low-rank matrix. Comput Adv Multi Sens Adapt Process CAMSAP 61(6):1–18Google Scholar
  32. 32.
    Liu DP, Li ZZ, Liu B, Chen WH, Liu TM, Cao L (2017) Infrared small target detection in heavy sky scene clutter based on sparse representation. Infrared Phys Technol 85:13–31CrossRefGoogle Scholar
  33. 33.
    Liu Q, Lai Z, Zhou Z, Kuang F, Jin Z (2016) A truncated nuclear norm regularization method based on weighted residual error for matrix completion. IEEE Trans Image Process 25(1):316–330MathSciNetCrossRefGoogle Scholar
  34. 34.
    Liu X, Zuo Z (2013) A dim small infrared moving target detection algorithm based on improved three-dimensional directional filtering. In: Proceedings of 2013 Chinese Conference on Image and Graphics Technologies. Springer, Berlin, pp 102–108Google Scholar
  35. 35.
    Liu Z, Li J, Li G, Bai J, Liu X (2017) A new model for sparse and low-rank matrix decomposition. J Appl Anal Comput 7(2):600–616MathSciNetGoogle Scholar
  36. 36.
    Ma S, Goldfarb D, Chen L (2011) Fixed point and Bregman iterative methods for matrix rank minimization. Math Program 128(1):321–353MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Otazo R, Candès E, Sodickson DK (2015) Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components. Magn Reson Med 73(3):1125–1136CrossRefGoogle Scholar
  38. 38.
    Oymak S, Mohan K, Fazel M, Hassibi B (2011) A simplified approach to recovery conditions for low rank matrices. In: Proceedings of 2011 IEEE international symposium on information theory (ISIT). IEEE, St. Petersburg, Russia, pp 2318–2322Google Scholar
  39. 39.
    Peng C, Kang Z, Yang M, Cheng Q (2016) RAP: Scalable RPCA for low-rank matrix recovery. In: Proceedings of the 25th ACM international conference on information and knowledge management. ACM, Indianapolis, USA, pp 2113–2118Google Scholar
  40. 40.
    Rahmani M, Atia GK (2015) Randomized subspace learning approach for high dimensional low rank plus sparse matrix decomposition. In: Proceedings of the 49th Asilomar conference on signals, systems and computers. IEEE, USA, pp 1796–1800CrossRefGoogle Scholar
  41. 41.
    Rahmani M, Atia G (2016) A subspace learning approach for high dimensional matrix decomposition with efficient column/row sampling. In: Proceedings of the 33rd international conference on machine learning, PMLR 48:1206–1214Google Scholar
  42. 42.
    Rahmani M, Atia GK (2017) High dimensional low rank plus sparse matrix decomposition. IEEE Trans Signal Process 65(8):2004–2019MathSciNetCrossRefGoogle Scholar
  43. 43.
    Ravishankar S, Moore BE, Nadakuditi RR, Fessler JA (2017) Low-rank and adaptive sparse signal (LASSI) models for highly accelerated dynamic imaging. IEEE Trans Med Imaging 36(5):1116–1128CrossRefGoogle Scholar
  44. 44.
    Recht B, Fazel M, Parrilo PA (2010) Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev 52(3):471–501MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Selesnick IW, Bayram I (2014) Sparse signal estimation by maximally sparse convex optimization. IEEE Trans Signal Process 62(5):1078–1092MathSciNetCrossRefGoogle Scholar
  46. 46.
    Shi C, Cheng Y, Wang J, Wang Y, Mori K, Tamura S (2017) Low-rank and sparse decomposition based shape model and probabilistic atlas for automatic pathological organ segmentation. Med Image Anal 38:30–49CrossRefGoogle Scholar
  47. 47.
    Shi W, Zhang X, Zou X, Han W, Min G (2017) Auditory mask estimation by RPCA for monaural speech enhancement. In: Proceedings of 2017 IEEE/ACIS 16th international conference on computer and information science (ICIS). IEEE, Wuhan, China, pp 179–184Google Scholar
  48. 48.
    Toh KC, Yun S (2010) An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems. Pac J Optim 6(3):615–640MathSciNetzbMATHGoogle Scholar
  49. 49.
    Vincent E, Gribonval R, F`evotte C (2006) Performance measurement in blind audio source separation. IEEE Trans Audio Speech Lang Process 14(4):1462–1469CrossRefGoogle Scholar
  50. 50.
    Wang Y, Su X (2014) Truncated nuclear norm minimization for image restoration based on iterative support detection. Math Prob Eng 2014:1–17MathSciNetGoogle Scholar
  51. 51.
    Werner R, Wilmsy M, Cheng B, Forkert ND (2016) Beyond cost function masking: RPCA-based non-linear registration in the context of VLSM. In: Proceedings of 2016 international workshop on pattern recognition in neuroimaging (PRNI). IEEE, Trento, Italy, pp 1–4Google Scholar
  52. 52.
    Xie Y, Gu S, Liu Y, Zuo W, Zhang W, Zhang L (2016) Weighted Schatten p-norm minimization for image denoising and background subtraction. IEEE Trans Image Process 25(10):4842–4857MathSciNetCrossRefGoogle Scholar
  53. 53.
    Yang AY, Sastry SS, Ganesh A, Ma Y (2010) Fast-minimization algorithms and an application in robust face recognition: a review. In: Proceedings of the 17th IEEE international conference on image processing. IEEE, Hong Kong, pp 1849–1852Google Scholar
  54. 54.
    Yu S, Zhang H, Duan Z (2017) Singing voice separation by low-rank and sparse spectrogram decomposition with prelearned dictionaries. J Audio Eng Soc 65(5):377–388CrossRefGoogle Scholar
  55. 55.
    Yuan X, Yang J (2009) Sparse and low-rank matrix decomposition via alternating direction methods. 12:1–16 (preprint) Google Scholar
  56. 56.
    Zheng CY, Li H (2013) Small infrared target detection based on low-rank and sparse matrix decomposition. Appl Mech Mater 239:214–218CrossRefGoogle Scholar
  57. 57.
    Zhou Z, Li X, Wright J, Candès EJ, Ma Y (2010) Stable principal component pursuit. In: Proceedings of the 2010 IEEE international symposium on information theory (ISIT). IEEE, Austin, Texas, pp 1518–1522Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics, College of SciencesChina Jiliang UniversityHangzhouPeople’s Republic of China

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