Skip to main content
Log in

On a Class of Orthonormal Algorithms for Principal and Minor Subspace Tracking

  • Published:
Journal of VLSI signal processing systems for signal, image and video technology Aims and scope Submit manuscript

Abstract

This paper elaborates on a new class of orthonormal power-based algorithms for fast estimation and tracking of the principal or minor subspace of a vector sequence. The proposed algorithms are closely related to the natural power method that has the fastest convergence rate among many power-based methods such as the Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method. A common feature of the proposed algorithms is the exact orthonormality of the weight matrix at each iteration. The orthonormality is implemented in a most efficient way. Besides the property of orthonormality, the new algorithms offer, as compared to other power based algorithms, a better numerical stability and a linear computational complexity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspace Methods for the Blind Identification of Multichannel FIR Filters,” IEEE Trans. Signal Processing, vol. 44,no. 2, 1995, pp. 516-525.

    Article  Google Scholar 

  2. X. Wang and V.H. Poor, “Blind Multiuser Detection: A Subspace Approach,” IEEE Trans. Inform. Theory, vol. 44,no. 2, 1998, pp. 677-690.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Chkeif, K. Abed-Meraim, G.K. Kaleh, and Y. Hua, “Spatio-Temporal Blind Adaptive Multiuser Detection,” IEEE Trans. Communication, vol. 48,no. 5, 2000, pp. 729-732.

    Article  Google Scholar 

  4. P. Comon and G.H. Golub, “Tracking a Few Extreme Singular Values and Vectors in Signal Processing,” Proc. of the IEEE, vol. 78,no. 8, 1990, pp. 1327-1343.

    Article  Google Scholar 

  5. B.H. Juang, S.Y. Kung, and C.A. Kamm, (Eds.), Proc. of the 1991 IEEE Workshop on Neural Networks for Signal Processing, Princeton, NJ, Sept. 30–Oct. 2, 1991.

  6. E. Oja, “A Simplified Neuron Model as a Principal Component Analyzer,” J. Math. Biology, vol. 15, 1982, pp. 267-273.

    Article  MathSciNet  MATH  Google Scholar 

  7. S.Y. Kung, K.I. Diamantaras, and J.S. Taur, “Adaptive Principal Component Extraction (APEX) and Applications,” IEEE Trans. on Signal Processing, vol. 42,no. 5, 1994, pp. 1202-1217.

    Article  Google Scholar 

  8. W.Y. Yan, U. Helmke, and J.B. Moore, “Global Analysis of Oja's Flow for Neural Networks,” IEEE Trans. on Neural Networks, vol. 5,no. 5, 1994, pp. 674-683.

    Article  Google Scholar 

  9. T. Chen, Y. Hua, and W. Yan, “Global Convergence of Oja's Subspace Algorithm for Principal Compoent Extraction,” IEEE Trans. on Neural Networks, vol. 9,no. 1, 1998, pp. 58-67.

    Article  Google Scholar 

  10. L. Xu, “Least Mean Square Error Reconstruction Principle for Self-Organizing Neural Nets,” Neural Networks, vol. 6, 1993, pp. 627-648.

    Article  Google Scholar 

  11. B. Yang, “Projection Approximation Subspace Tracking,” IEEE Trans. Signal Processing, vol. 44,no. 1, 1995, pp. 95-107.

    Article  Google Scholar 

  12. Y. Miao and Y. Hua, “Fast Subspace Tracking and Neural Network Learning by a Novel Information Criterion,” IEEE Trans. on Signal Processing, vol. 46,no. 7, 1998, pp. 1967-1979.

    Article  Google Scholar 

  13. Y. Hua, Y. Xiang, T. Chen, K. Abed-Meraim, and Y. Miao, “A New Look at the Power Method for Fast Subspace Tracking,” Digital Signal Processing, Academic Press (Short version is available in Proc. of IEEE Workshop on Neural Networks for Signal Processing, April 1999), Oct. 1999.

  14. G.H. Golub and C.F. Van Loan, Matrix Computations, Baltimore, MD: Johns Hopkins University Press, 1989.

    MATH  Google Scholar 

  15. G.W. Stewart, “An Updating Algorithm for Subspace Tracking,” IEEE Trans. on Signal Processing, vol. 40,no. 6, 1992, pp. 1535-1541.

    Article  Google Scholar 

  16. C.S. MacInnes, “Fast, Accurate Subspace Tracking Using Operator Restriction Analysis,” in Proc. of IEEE ICASSP, 1998, pp. 1357-1360.

  17. C. Riou and T. Chonavel, “Fast Adaptive Eigenvalue Decomposition: A Maximum Likelihood Approach,” in Proc. of IEEE ICASSP'97, (Munich) Germany, 1997, pp. 3565-3568.

  18. T. Gustafsson and C.S. MacInnes, “A Class of Subspace Tracking Algorithms Based on Approximation of the Noise Subspace,” IEEE Tr. on Signal Processing, vol. 48,no. 11, 2000, pp. 3231-3235.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Srivastava, “A Bayesian Approach to Geometric Subspace Estimation,” IEEE Tr. on Signal Processing, vol. 48,no. 5, 2000, pp. 1390-1400.

    Article  Google Scholar 

  20. S. Marcos, A. Marsal, and M. Benidir, “The Propagator Method for Source Bearing Estimation,” Signal Processing, vol. 42, 1995, pp. 121-138.

    Article  Google Scholar 

  21. K. Abed-Meraim, S. Attallah, A. Chkeif, and Y. Hua, “Orthonormal Oja Algorithm,” IEEE Signal Processing Letters, vol. 7,no. 5, 2000.

  22. C.B. Papadias, “A Globally Convergent Algorithm for Blind Source Separation,” in Proc. EUSIPCO, Tampere, Finland, Sept. 2000.

  23. S.C. Douglas, “Self-Stabilized Gradient Algorithms for Blind Source Separation with Orthogonality Constraints,” IEEE Tr. on Neural Networks, vol. 11,no. 6, 2000, pp. 1490-1497.

    Article  Google Scholar 

  24. S.C. Douglas, “Numerically-Robust Adaptive Subspace Tracking Using Householder Transformations,” in Proc. of Asilomar Conference, Nov. 2000, pp. 499-503.

  25. Y. Hua, T. Chen, and Y. Miao, “A Unifying View of a Class of Subspace Tracking Methods,” in ISSPR'98, Hong Kong, vol. II, Sept. 1998, pp. 27-32.

    Google Scholar 

  26. K. Abed-Meraim, A. Chkeif, and Y. Hua, “Fast Orthonormal PAST Algorithm,” IEEE Signal Processing Letters, vol. 7,no. 3, 2000, pp. 60-62.

    Article  Google Scholar 

  27. S. Attallah and K. Abed-Meraim, “Fast Algorithms for Subspace Tracking,” IEEE Signal Processing Letters, vol. 8,no. 7, 2001, pp. 203-206.

    Article  Google Scholar 

  28. T. Chen, S.-I. Amari, and Q. Lin, “A Unified Algorithm for Principal and Minor Components Extraction,” Neural Networks, vol. 11, 1998, pp. 385-390.

    Article  Google Scholar 

  29. S.C. Douglas, S.-Y. Kung, and S.-I. Amari, “A Self-Stabilized Minor Subspace Rule,” Signal Processing Letters, vol. 5,no. 12, Dec. 1998, pp. 328-330.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abed-Meraim, K., Chkeif, A., Hua, Y. et al. On a Class of Orthonormal Algorithms for Principal and Minor Subspace Tracking. The Journal of VLSI Signal Processing-Systems for Signal, Image, and Video Technology 31, 57–70 (2002). https://doi.org/10.1023/A:1014445221814

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014445221814

Navigation