Robust Adaptive Modified Newton Algorithm for Generalized Eigendecomposition and Its Application

  • Jian Yang
  • Feng Yang
  • Hong-Sheng Xi
  • Wei Guo
  • Yanmin Sheng
Open Access
Research Article
Part of the following topical collections:
  1. Numerical Linear Algebra in Signal Processing Applications


We propose a robust adaptive algorithm for generalized eigendecomposition problems that arise in modern signal processing applications. To that extent, the generalized eigendecomposition problem is reinterpreted as an unconstrained nonlinear optimization problem. Starting from the proposed cost function and making use of an approximation of the Hessian matrix, a robust modified Newton algorithm is derived. A rigorous analysis of its convergence properties is presented by using stochastic approximation theory. We also apply this theory to solve the signal reception problem of multicarrier DS-CDMA to illustrate its practical application. The simulation results show that the proposed algorithm has fast convergence and excellent tracking capability, which are important in a practical time-varying communication environment.


Nonlinear Optimization Hessian Matrix Stochastic Approximation Adaptive Modify Nonlinear Optimization Problem 
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  1. 1.
    Lu J, Plataniotis KN, Venetsanopoulos AN: Face recognition using LDA-based algorithms. IEEE Transactions on Neural Networks 2003,14(1):195-200. 10.1109/TNN.2002.806647CrossRefGoogle Scholar
  2. 2.
    Fidler S, Skočaj D, Leonardis A: Combining reconstructive and discriminative subspace methods for robust classification and regression by subsampling. IEEE Transactions on Pattern Analysis and Machine Intelligence 2006,28(3):337-350.CrossRefGoogle Scholar
  3. 3.
    Wong TF, Lok TM, Lehnert JS, Zoltowski MD: A linear receiver for direct-sequence spread-spectrum multiple-access systems with antenna arrays and blind adaptation. IEEE Transactions on Information Theory 1998,44(2):659-676. 10.1109/18.661511MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Yang J, Xi H, Yang F, Zhao Y: Fast adaptive blind beamforming algorithm for antenna array in CDMA systems. IEEE Transactions on Vehicular Technology 2006,55(2):549-558. 10.1109/TVT.2005.863419CrossRefGoogle Scholar
  5. 5.
    Yang B: Projection approximation subspace tracking. IEEE Transactions on Signal Processing 1995,43(1):95-107. 10.1109/78.365290CrossRefGoogle Scholar
  6. 6.
    Ouyang S, Ching PC, Lee T: Robust adaptive quasi-Newton algorithms for eigensubspace estimation. IEE Proceedings: Vision, Image and Signal Processing 2003,150(5):321-330. 10.1049/ip-vis:20030767Google Scholar
  7. 7.
    Hyvärinen A, Karhunen J, Oja E: Independent Component Analysis. John Wiley & Sons, New York, NY, USA; 2001.CrossRefGoogle Scholar
  8. 8.
    Golub GH, VanLoan CF: Matrix Computations. John Hopkins University Press, Baltimore, Md, USA; 1991.Google Scholar
  9. 9.
    Mathew G, Reddy VU: A quasi-Newton adaptive algorithm for generalized symmetric eigenvalue problem. IEEE Transactions on Signal Processing 1996,44(10):2413-2422. 10.1109/78.539027CrossRefGoogle Scholar
  10. 10.
    Chatterjee C, Roychowdhury VP, Ramos J, Zoltowski MD: Self-organizing algorithms for generalized eigen-decomposition. IEEE Transactions on Neural Networks 1997,8(6):1518-1530. 10.1109/72.641473CrossRefGoogle Scholar
  11. 11.
    Xu D, Principe JC, Wu H-C: Generalized eigendecomposition with an on-line local algorithm. IEEE Signal Processing Letters 1998,5(11):298-301. 10.1109/97.728475CrossRefGoogle Scholar
  12. 12.
    Morgan DR: Adaptive algorithms for solving generalized eigenvalue signal enhancement problems. Signal Processing 2004,84(6):957-968. 10.1016/j.sigpro.2004.02.002CrossRefMATHGoogle Scholar
  13. 13.
    Rao YN, Principe JC, Wong TF: Fast RLS-like algorithm for generalized eigendecomposition and its applications. The Journal of VLSI Signal Processing 2004,37(2-3):333-344.CrossRefMATHGoogle Scholar
  14. 14.
    Yang J, Xi H, Yang F, Zhao Y: RLS-based adaptive algorithms for generalized eigen-decomposition. IEEE Transactions on Signal Processing 2006,54(4):1177-1188.CrossRefGoogle Scholar
  15. 15.
    Lok TM, Wong TF, Lehnert JS: Blind adaptive signal reception for MC-CDMA systems in Rayleigh fading channels. IEEE Transactions on Communications 1999,47(3):464-471. 10.1109/26.752827CrossRefGoogle Scholar
  16. 16.
    Kondo S, Milstein LB: Performance of multicarrier DS CDNA systems. IEEE Transactions on Communications 1996,44(2):238-246. 10.1109/26.486616CrossRefGoogle Scholar
  17. 17.
    Proakis JG: Digital Communications. McGraw-Hill, New York, NY, USA; 1995.MATHGoogle Scholar
  18. 18.
    Namgoong J, Wong TF, Lehnert JS: Subspace multiuser detection for multicarrier DS-CDMA. IEEE Transactions on Communications 2000,48(11):1897-1908. 10.1109/26.886487CrossRefGoogle Scholar
  19. 19.
    Haykin S: Adaptive Filter Theory. Prentice-Hall, Upper Saddle River, NJ, USA; 2002.MATHGoogle Scholar
  20. 20.
    Ljung L: Analysis of recursive stochastic algorithms. IEEE Transactions on Automatic Control 1977,22(4):551-575. 10.1109/TAC.1977.1101561MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kushner HJ, Clark DS: Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer, New York, NY, USA; 1978.CrossRefMATHGoogle Scholar
  22. 22.
    Morgan DR, Benesty J, Sondhi MM: On the evaluation of estimated impulse responses. IEEE Signal Processing Letters 1998,5(7):174-176. 10.1109/97.700920CrossRefGoogle Scholar
  23. 23.
    Lok TM, Wong TF: Transmitter and receiver optimization in multicarrier CDMA systems. IEEE Transactions on Communications 2000,48(7):1197-1207. 10.1109/26.855527CrossRefGoogle Scholar

Copyright information

© Jian Yang et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Jian Yang
    • 1
  • Feng Yang
    • 1
  • Hong-Sheng Xi
    • 1
  • Wei Guo
    • 1
  • Yanmin Sheng
    • 1
  1. 1.Laboratory of Network Communication System and Control, Department of AutomationUniversity of Science and Technology of ChinaHefeiChina

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