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Bayesian and geometric subspace tracking

Part of: Probabilistic methods, simulation and stochastic differential equations Communication, information Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Anuj Srivastava  and
Eric Klassen
Anuj Srivastava*
Affiliation:
Florida State University
Eric Klassen*
Affiliation:
Florida State University
*
∗ Postal address: Department of Statistics, Florida State University, Tallahassee, FL 32306, USA. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA.
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Abstract

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.

Keywords

Subspace trackingGrassmann manifoldBayesian trackingnon-Euclidean filteringsequential Monte Carlo

MSC classification

Primary: 93E11: Filtering
Secondary: 94A11: Application of orthogonal and other special functions 65C35: Stochastic particle methods
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 43 - 56
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Bucy, R. S. (1991). Geometry and multiple direction estimation. Inf. Sci. 57/58, 145–158.Google Scholar
[2] Comon, P. (1994). Independent component analysis, a new concept? Signal Process. 36, 287–314.Google Scholar
[3] Darling, R. (2002). Intrinsic location parameter of a diffusion process. Electron. J. Prob. 7, No. 3.Google Scholar
[4] Delmas, J. P. and Cardoso, J. F. (1998). Performance analysis of an adaptive algorithm for tracking dominant subspaces. IEEE Trans. Signal Process. 46, 3045–3057.Google Scholar
[5] Doucet, E. A., de Freitas, N. and Gordon, N. (eds) (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.Google Scholar
[6] Edelman, A., Arias, T. and Smith, S. T. (1998). The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20, 303–353.Google Scholar
[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. 301–304.Google Scholar
[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, 107–113.Google Scholar
[9] Helgason, S. (1978). Differential Geometry, Lie Groups and Symmetric Spaces (Pure Appl. Math. 80). Academic Press, New York.Google Scholar
[10] Jolliffe, I. T. (1986). Principal Component Analysis. Springer, New York.Google Scholar
[11] Jupp, P. E. and Kent, J. T. (1987). Fitting smooth paths to spherical data. Appl. Statist. 36, 34–46.Google Scholar
[12] Karcher, H. (1977). Riemann center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541.Google Scholar
[13] Kendall, W. S. (1990). Probability, convexity, and harmonic maps with small image I. Uniqueness and fine existence. Proc. London Math. Soc. 61, 371–406.Google Scholar
[14] Kobayashi, S. and Nomizu, K. (1969). Foundations of Differential Geometry, Vol. 2. Interscience Publishers, New York.Google Scholar
[15] Le, H. (1991). On geodesics in Euclidean shape spaces. J. London Math. Soc. 44, 360–372.Google Scholar
[16] Le, H. (2001). Locating Fréchet means with application to shape spaces. Adv. Appl. Prob. 33, 324–338.Google Scholar
[17] Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93, 1032–1044.Google Scholar
[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, 2678–2690.Google Scholar
[19] Oller, J. M. and Corcuera, J. M. (1995). Intrinsic analysis of statistical estimation. Ann. Statist. 23, 1562–1581.Google Scholar
[20] Schmidt, R. (1981). A signal subspace approach to multiple emitter location and spectral estimation. Doctoral Thesis, Stanford University.Google Scholar
[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. 793–797.Google Scholar
[22] Srivastava, A. (2000). A Bayesian approach to geometric subspace estimation. IEEE Trans. Signal Process. 48, 1390–1400.Google Scholar
[23] Srivastava, A. and Klassen, E. (2001). Monte Carlo extrinsic estimators for manifold-valued parameters. IEEE Trans. Signal Process. 50, 299–308.Google Scholar
[24] Tong, L. and Perreau, S. (1998). Multichannel blind estimation: from subspace to maximum likelihood methods. Proc. IEEE 86, 1951–1968.Google Scholar
[25] Warner, F. W. (1994). Foundations of Differentiable Manifolds and Lie Groups. Springer, New York.Google Scholar