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Robust Adaptive Modified Newton Algorithm for Generalized Eigendecomposition and Its Application


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.


  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.2002.806647

    Article  Google 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.

    Article  Google 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. 10.1109/18.661511

    MathSciNet  Article  Google 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.2005.863419

    Article  Google Scholar 

  5. 5.

    Yang B: Projection approximation subspace tracking. IEEE Transactions on Signal Processing 1995,43(1):95-107. 10.1109/78.365290

    Article  Google 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:20030767

    Google Scholar 

  7. 7.

    Hyvärinen A, Karhunen J, Oja E: Independent Component Analysis. John Wiley & Sons, New York, NY, USA; 2001.

    Book  Google 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. 10.1109/78.539027

    Article  Google 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. 10.1109/72.641473

    Article  Google 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. 10.1109/97.728475

    Article  Google 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.2004.02.002

    Article  Google 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.

    Article  Google 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.

    Article  Google 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. 10.1109/26.752827

    Article  Google Scholar 

  16. 16.

    Kondo S, Milstein LB: Performance of multicarrier DS CDNA systems. IEEE Transactions on Communications 1996,44(2):238-246. 10.1109/26.486616

    Article  Google Scholar 

  17. 17.

    Proakis JG: Digital Communications. McGraw-Hill, New York, NY, USA; 1995.

    MATH  Google 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. 10.1109/26.886487

    Article  Google Scholar 

  19. 19.

    Haykin S: Adaptive Filter Theory. Prentice-Hall, Upper Saddle River, NJ, USA; 2002.

    MATH  Google Scholar 

  20. 20.

    Ljung L: Analysis of recursive stochastic algorithms. IEEE Transactions on Automatic Control 1977,22(4):551-575. 10.1109/TAC.1977.1101561

    MathSciNet  Article  Google Scholar 

  21. 21.

    Kushner HJ, Clark DS: Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer, New York, NY, USA; 1978.

    Book  Google 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. 10.1109/97.700920

    Article  Google Scholar 

  23. 23.

    Lok TM, Wong TF: Transmitter and receiver optimization in multicarrier CDMA systems. IEEE Transactions on Communications 2000,48(7):1197-1207. 10.1109/26.855527

    Article  Google Scholar 

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Correspondence to Jian Yang.

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Yang, J., Yang, F., Xi, HS. et al. Robust Adaptive Modified Newton Algorithm for Generalized Eigendecomposition and Its Application. EURASIP J. Adv. Signal Process. 2007, 038341 (2007).

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  • Nonlinear Optimization
  • Hessian Matrix
  • Stochastic Approximation
  • Adaptive Modify
  • Nonlinear Optimization Problem