Neural Computing and Applications

, Volume 29, Issue 5, pp 1455–1463 | Cite as

Simplified neural network for generalized least absolute deviation

Original Article
First Online:
Received:
Accepted:
  • 116 Downloads

Abstract

This paper proposes a simplified neural network for generalized least absolute deviation by transforming its optimization conditions into a system of double projection equations. The proposed network is proved to be stable in the sense of Lyapunov and converges to an exact optimization solution of the original problem for any starting point. Compared with the existing neural networks for generalized least absolute deviation, the new model has the least neurons and low complexity and is suitable to parallel implementation. The validity and transient behavior of the proposed neural network are demonstrated by numerical examples.

Keywords

Neural network Generalized least absolute deviation Convergence Stability 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The authors would like to thank the Editor-in-Chief and three anonymous reviewers for their insightful and constructive comments, which have enriched the content and improved the presentation of this paper. This work was supported in part by the National Natural Science Foundation of China under Grant 61273311, and the Fundamental Research Funds for the Central Universities under Grant GK201603002.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests (either financial or nonfinancial) regarding the publication of the paper.

References

  1. 1.
    Xia Y, Kamel M (2008) A generalized least absolute deviation method for parameter estimation of autoregressive signals. IEEE Trans Neural Netw 19(1):107–118CrossRefGoogle Scholar
  2. 2.
    Bloomfield P, Steiger W (1983) Least absolute deviations: theory applications and algorithms. Brikhäuser, BostonMATHGoogle Scholar
  3. 3.
    Shi M, Lukas M (2002) An L 1 estimation algorithm with degeneracy and linear constraints. Comput Stat Data Anal 39(1):35–55MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Liu X, Gao X et al (2012) Image restoration method based on l 1-nonconvex nonsmooth function. Comput Eng 38:190–193Google Scholar
  5. 5.
    Bazaraa M, Sherali H, Shetty C (1993) Nonlinear programming: theory and algorithms, 2nd edn. Wiley, New YorkMATHGoogle Scholar
  6. 6.
    Xia Y, Kamel M (2007) Cooperative recurrent neural networks for the constrained L 1 estimator. IEEE Trans Signal Process 55(7):3192–3206MathSciNetCrossRefGoogle Scholar
  7. 7.
    Wang Z, Peterson B (2008) Constrained least absolute deviation neural networks. IEEE Trans Neural Netw 19(2):273–283CrossRefGoogle Scholar
  8. 8.
    Xia Y (2009) A compact cooperative recurrent neural network for computing general constrained L 1 norm estimators. IEEE Trans Signal Process 57(9):3693–3697MathSciNetCrossRefGoogle Scholar
  9. 9.
    Wang Z, He Z, Chen J (2005) Robust time delay estimation of bioelectric signals. IEEE Trans Bio med Eng 52(3):454–462CrossRefGoogle Scholar
  10. 10.
    Hu X, Sun C, Zhang B (2010) Design of recurrent neural networks for solving constrained least absolute deviation problems. IEEE Trans Neural Netw 21(7):1073–1086CrossRefGoogle Scholar
  11. 11.
    Xia Y (2010) A fast algorithm for constrained GLAD estimation with application to image restoration. In: Proceedings of the 8th world congress on intelligent control and automation, Jinan, China, JulyGoogle Scholar
  12. 12.
    Hopield J, Tank D (1986) Computing with neural circuits: a model. Science 233(4764):625–633CrossRefGoogle Scholar
  13. 13.
    Tank D, Hopfield J (1986) Simple neural optimization networks: an a/d converter, signal decision circuit, and a linear programming circuit. IEEE Trans Circuits Syst 33:533–541CrossRefGoogle Scholar
  14. 14.
    Kennedy M, Chua L (1988) Neural networks for nonlinear programming. IEEE Trans Circuits Syst 35:554–562MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zhang S, Constantinides A (1992) Langrange programming neural networks. IEEE Trans Circuits Syst 39(7):441–452CrossRefMATHGoogle Scholar
  16. 16.
    Xue X, Bian W (2008) Subgradient-based neural networks for nonsmooth convex optimization problems. IEEE Trans Circuits Syst I Regul Pap 55(8):2378–2391MathSciNetCrossRefGoogle Scholar
  17. 17.
    Qin S, Xue X (2015) A two-layer recurrent neural network for nonsmooth convex optimization problems. IEEE Trans Neural Netw Learn Syst 26(6):1149–1160MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bian W, Xue X (2013) Neural network for solving constrained convex optimization problems with global attractivity. IEEE Trans Circuits Syst I Regul Pap 60(3):710–723MathSciNetCrossRefGoogle Scholar
  19. 19.
    Qin S et al (2015) Neural network for constrained nonsmooth optimization using Tikhonov regularization. Neural Netw 63:272–281CrossRefMATHGoogle Scholar
  20. 20.
    Liu X, Zhou M (2016) A one-layer recurrent neural network for non-smooth convex optimization subject to linear inequality constraints. Chaos Solitons Fractals 87:39–46MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Wang Z, Cheung J et al (2000) Neural implementation of unconstrained minimum L 1-norm optimization least absolute deviation model and its application to time delay estimation. IEEE Trans Circuits Syst II Analog Digital Sig Process 47(11):1214–1226CrossRefGoogle Scholar
  22. 22.
    Xia Y, Kamel M (2008) A cooperative recurrent neural network for solving L 1 estimation problems with general linear constraints. Neural Comput 20(3):844–872MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Liu Q, Zhao Y, Cheng L (2015) Continuous-time multi-agent network for distributed least absolute deviation. In: International Symposium on Neural Networks. Springer International Publishing, pp 436–443Google Scholar
  24. 24.
    Liu Q, Wang J (2015) A second-order multi-agent network for bound-constrained distributed optimization. IEEE Trans Autom Control 60(12):3310–3315MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Liu Q, Wang J (2016) L 1-minimization algorithms for sparse signal reconstruction based on a projection neural network. IEEE Trans Neural Netw Learn Syst 27(3):698–707MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li C, Gao X, Li Y, Liu R (2016) A new neural network for l 1-norm programming. Neurocomputing 202:98–103CrossRefGoogle Scholar
  27. 27.
    Xia Y, Kamel M (2007) Novel cooperative neural fusion algorithms for image restoration and image fusion. IEEE Trans Image Process 16(2):367–381MathSciNetCrossRefGoogle Scholar
  28. 28.
    Xia Y, Sun C, Zheng W (2012) Discrete-time neural network for fast solving large linear L 1 estimation problems and its application to image restoration. IEEE Trans Neural Netw Learn Syst 23(5):812–820CrossRefGoogle Scholar
  29. 29.
    Gao X (2001) A neural network for a class of extended linear variational inequalities. Chin J Electron 10(4):471–475Google Scholar
  30. 30.
    Gao X (2004) A novel neural network for nonlinear convex programming. IEEE Trans Neural Netw 15(3):613–621CrossRefGoogle Scholar
  31. 31.
    Gao X, Du L (2006) A neural network with finite-time convergence for a class of variational inequalities. Lect Notes Comput Sci 4113:32–41CrossRefGoogle Scholar
  32. 32.
    Gao X, Liao L (2003) A neural network for monotone variational inequalities with linear constraints. Phys Lett A 307(2–3):118–128MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Gao X, Liao L (2006) A novel neural network for a class of convex quadratic minimax problems. Neural Comput 18(8):1818–1846MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Gao X, Liao L (2009) A new projection-based neural network for constrained variational inequalities. IEEE Trans Neural Netw 15(4):622–628Google Scholar
  35. 35.
    Gao X, Liao L (2010) A new one-layer network for linear and quadratic programming. IEEE Trans Neural Netw 21(6):918–929CrossRefGoogle Scholar
  36. 36.
    He X, Huang T et al (2017) An inertial projection neural network for solving variational inequalities. IEEE Trans Cybern 47(3):809–814CrossRefGoogle Scholar
  37. 37.
    Liu Q, Wang J (2015) A projection neural network for constrained quadratic minimax optimization. IEEE Trans Neural Netw Learn Syst 26(11):2891–2900MathSciNetCrossRefGoogle Scholar
  38. 38.
    Friesz T, Bernstein D et al (1994) Day-to-day dynamic network disequilibria and idealized traveler information systems. Oper Res 42:1120–1136MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Xia Y, Wang J (1998) Neural networks for solving least absolute and related problems. Neurocomputing 19:13–21CrossRefGoogle Scholar
  40. 40.
    Wang Z, Cheung J, Xia Y (2000) Minimum fuel neural networks and their applications to overcomplete signal respresentations. IEEE Trans Circuit Syst I Fundam Theory Appl 47(8):1146–1159CrossRefMATHGoogle Scholar
  41. 41.
    Li G, Yan Z, Wang J (2015) A one-layer recurrent neural network for constrained nonconvex optimization. Neural Netw 61:10–21CrossRefMATHGoogle Scholar
  42. 42.
    Li G, Yan Z, Wang J (2014) A one-layer recurrent neural network for constrained nonsmooth invex optimization. Neural Netw 50:79–89CrossRefMATHGoogle Scholar
  43. 43.
    Liu Q, Dang C, Huang T (2013) A one-layer recurrent neural network for real-time portfolio optimization with probability criterion. IEEE Trans Cybern 43(1):12–23Google Scholar
  44. 44.
    Li S, Li Y, Wang Z (2013) A class of finite-time dual neural networks for solving quadratic programming problems and its k-winners-take-all application. Neural Netw Off J Int Neural Netw Soc 39(1):27–39CrossRefMATHGoogle Scholar
  45. 45.
    Li S, Chen S, Liu B (2013) Accelerating a recurrent neural network to finite-time convergence for solving time-varying sylvester equation by using a sign-bi-power activation function. Neural Process Lett 37(2):189–205CrossRefGoogle Scholar
  46. 46.
    Li S, Li Y (2014) Nonlinearly activated neural network for solving time-varying complex sylvester equation. IEEE Trans Cybern 44(8):1397–1407CrossRefGoogle Scholar
  47. 47.
    Li S, He J et al (2017) Distributed recurrent neural networks for cooperative control of manipulators: a game-theoretic perspective. IEEE Trans Neural Netw Learn Syst 28(2):415–426MathSciNetCrossRefGoogle Scholar
  48. 48.
    Li S, Zhang Y, Jin L (2016) Kinematic control of redundant manipulators using neural networks. IEEE Trans Neural Netw Learn Syst. doi: 10.1109/TNNLS.2016.2574363 MathSciNetGoogle Scholar
  49. 49.
    Liu S, Wang J (2006) A simplified dual neural network for quadratic programming with its kwta application. IEEE Trans Neural Netw 17(6):1500–1510CrossRefGoogle Scholar
  50. 50.
    Kinderlehrer D, Stampcchia G (1980) An introduction to variational inequalities and their applications. Academic, New YorkGoogle Scholar
  51. 51.
    La Salle J (1976) The stability of dynamical systems. SIAM, PhiladelphiaCrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2017

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China

Personalised recommendations