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Bayesian and geometric subspace tracking

  • Anuj Srivastava (a1) and Eric Klassen (a1)


We address the problem of tracking the time-varying linear subspaces (of a larger system) under a Bayesian framework. Variations in subspaces are treated as a piecewise-geodesic process on a complex Grassmann manifold and a Markov prior is imposed on it. This prior model, together with an observation model, gives rise to a hidden Markov model on a Grassmann manifold, and admits Bayesian inferences. A sequential Monte Carlo method is used for sampling from the time-varying posterior and the samples are used to estimate the underlying process. Simulation results are presented for principal subspace tracking in array signal processing.


Corresponding author

Postal address: Department of Statistics, Florida State University, Tallahassee, FL 32306, USA. Email address:
∗∗ Postal address: Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA.


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[1] Bucy, R. S. (1991). Geometry and multiple direction estimation. Inf. Sci. 57/58, 145158.
[2] Comon, P. (1994). Independent component analysis, a new concept? Signal Process. 36, 287314.
[3] Darling, R. (2002). Intrinsic location parameter of a diffusion process. Electron. J. Prob. 7, No. 3.
[4] Delmas, J. P. and Cardoso, J. F. (1998). Performance analysis of an adaptive algorithm for tracking dominant subspaces. IEEE Trans. Signal Process. 46, 30453057.
[5] Doucet, E. A., de Freitas, N. and Gordon, N. (eds) (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
[6] Edelman, A., Arias, T. and Smith, S. T. (1998). The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20, 303353.
[7] Gallivan, K., Srivastava, A. and Van Dooren, P. (2003). Efficient algorithms for inferences on Grassmann manifolds. In Proc. 12th IEEE Workshop Statist. Signal Processing (Saint Louis, MO, September–October 2003), IEEE, Los Alamitos, CA, pp. 301304.
[8] Gordon, N. J., Salmon, D. J. and Smith, A. F. M. (1993). A novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEEE Proc. Radar Signal Process. 140, 107113.
[9] Helgason, S. (1978). Differential Geometry, Lie Groups and Symmetric Spaces (Pure Appl. Math. 80). Academic Press, New York.
[10] Jolliffe, I. T. (1986). Principal Component Analysis. Springer, New York.
[11] Jupp, P. E. and Kent, J. T. (1987). Fitting smooth paths to spherical data. Appl. Statist. 36, 3446.
[12] Karcher, H. (1977). Riemann center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509541.
[13] Kendall, W. S. (1990). Probability, convexity, and harmonic maps with small image I. Uniqueness and fine existence. Proc. London Math. Soc. 61, 371406.
[14] Kobayashi, S. and Nomizu, K. (1969). Foundations of Differential Geometry, Vol. 2. Interscience Publishers, New York.
[15] Le, H. (1991). On geodesics in Euclidean shape spaces. J. London Math. Soc. 44, 360372.
[16] Le, H. (2001). Locating Fréchet means with application to shape spaces. Adv. Appl. Prob. 33, 324338.
[17] Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93, 10321044.
[18] Miller, M. I., Srivastava, A. and Grenander, U. (1995). Conditional-expectation estimation via jump-diffusion processes in multiple target tracking/recognition. IEEE Trans. Signal Process. 43, 26782690.
[19] Oller, J. M. and Corcuera, J. M. (1995). Intrinsic analysis of statistical estimation. Ann. Statist. 23, 15621581.
[20] Schmidt, R. (1981). A signal subspace approach to multiple emitter location and spectral estimation. Doctoral Thesis, Stanford University.
[21] Smith, S. T. (1997). Subspace tracking with full rank updates. In Proc. 31st Asilomar Conf. Signals, Systems and Computers (Monterey, CA, November 1997), IEEE, Los Alamitos, CA, pp. 793797.
[22] Srivastava, A. (2000). A Bayesian approach to geometric subspace estimation. IEEE Trans. Signal Process. 48, 13901400.
[23] Srivastava, A. and Klassen, E. (2001). Monte Carlo extrinsic estimators for manifold-valued parameters. IEEE Trans. Signal Process. 50, 299308.
[24] Tong, L. and Perreau, S. (1998). Multichannel blind estimation: from subspace to maximum likelihood methods. Proc. IEEE 86, 19511968.
[25] Warner, F. W. (1994). Foundations of Differentiable Manifolds and Lie Groups. Springer, New York.


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Bayesian and geometric subspace tracking

  • Anuj Srivastava (a1) and Eric Klassen (a1)


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