The modified natural power method for principal component computation

Abstract

A modified version of the natural power method (NP) for fast estimation and tracking of the principal eigenvectors of a vector sequence is Presented. It is an extension of the natural power method because it is a solution to obtain the principal eigenvectors and not only for tracking of the principal subspace. As compared with some power-based methods such as Oja method, the projection approximation subspace tracking (PAST) method, and the novel information criterion (NIC) method, the modified natural power method (MNP) has the fastest convergence rate and can be easily implemented with only O(np) flops of computation at each iteration, where n is the dimension of the vector sequence and p is the dimension of the principal subspace or the number of the principal eigenvectors. Furthermore, it is guaranteed to be globally and exponentially convergent in contrast with some non-power-based methods such as MALASE and OPERA.

Keywords

PCA / MCA / power-based / exponentially convergent / 93B40 / 93C05

References

[1] Comon P., Golub G.H. Tracking a few extreme singular values and vectors in signal processing. Proc of the IEEE, 1990, 78(8): 1327-1343.

[2] Oja E. A simplified neuron model as a principal component analyzer. J. Math. Biology, 1982, 15: 267-273.

[3] Kung S. Y., Diamantaras K. I., Taur J. S. Adaptive principal component extraction (APEX) and applications. IEEE Trans on Signal Processing, 1994, 42(5): 1202-1217.

[4] Yan W. Y., Helmke U., Moore J. B. Global analysis of Oja’s flow for neural networks. IEEE Trans on Neural Networks, 1994, 5(5): 674-683.

[5] Chen T., Hua Y., Yan W. Global convergence of Oja’s subspace algorithm for principal component extraction. IEEE Trans on Neural Networks, 1998, 9(1): 58-67.

[6] Xu L. Least mean square error reconstruction principle for self-organizing neural nets. Neural Networks, 1993, 6: 627-648.

[7] Yang B. Projection approximation subspace tracking. IEEE Trans on Signal Processing, 1995, 43(1): 95-107.

[8] Miao Y., Hua Y. Fast subspace tracking and neural network learning by a novel information criterion. IEEE Trans on Signal Processing, 1998, 46(7): 1967-1979.

[9] Golub G. H., Van Loan C. F. Matrix Computations, 1989, Baltimore, MD: Johns Hopkins Univ Press.

[10] Hua Y., Xiang Y., Chen T., Abed-Meraim K. and Miao Y., 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

[11] Riou C. and Chonavel T., Fast adaptive eigenvalue decomposition: a maximum likelihood approach, Proc. of IEEE ICASSP’97, 1997: 3565–3568

[12] MacInnes C. S., Fast accurate subspace tracking using operator restriction analysis, Proc. of IEEE ICASSP’98, 1998: 1357–1360

[13] Chen T. P., Amari S., Lin Q. A Unified approach for Principal and Minor Components Extraction. Neural Networks, 1998, 11: 385-390.

[14] Yang B. Projection approximation subspace tracking. IEEE Trans on Signal Processing, 1995, 43(1): 95-107.