Geometric Filtering for Subspace Tracking

We address the problem of tracking principal subspaces using ideas from nonlinear ltering. The subspaces are represented by their complex projection-matrices, and time-varying subspaces correspond to trajectories on the Grassmann manifold. Under a Bayesian approach, we impose a smooth prior on the velocities associated with the subspace motion. This prior combined with any standard likelihood function forms a posterior density on the Grassmannian, for ltering and estimation. Using a sequential Monte Carlo method, a recursive nonlinear tracking algorithm is derived and some implementation results are presented.

[1]  Michael Isard,et al.  Active Contours , 2000, Springer London.

[2]  U. Grenander,et al.  nal-Mean Estimation Via Jump-Diffusion ses in Multiple Target Tracking/Recognition , 1995 .

[3]  Donald W. Tufts,et al.  Two algorithms for fast approximate subspace tracking , 1999, IEEE Trans. Signal Process..

[4]  Jun S. Liu,et al.  Metropolized independent sampling with comparisons to rejection sampling and importance sampling , 1996, Stat. Comput..

[5]  Y. Hua,et al.  Fast orthonormal PAST algorithm , 2000, IEEE Signal Processing Letters.

[6]  Karim Abed-Meraim,et al.  A New Look at the Power Method for Fast Subspace Tracking , 1999, Digit. Signal Process..

[7]  L. Tong,et al.  Multichannel blind identification: from subspace to maximum likelihood methods , 1998, Proc. IEEE.

[8]  Y. Bar-Shalom.  Tracking and data association , 1988 .

[9]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[10]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[11]  Richard S. Bucy,et al.  Geometry and multiple direction estimation , 1991, Inf. Sci..

[12]  Benoît Champagne,et al.  On the efficient use of Givens rotations in SVD-based subspace tracking algorithms , 1999, Signal Process..

[13]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[14]  Anuj Srivastava,et al.  Subspace tracking via rigid body dynamics , 1996, Proceedings of 8th Workshop on Statistical Signal and Array Processing.

[15]  Jean Pierre Delmas,et al.  Performance analysis of an adaptive algorithm for tracking dominant subspaces , 1998, IEEE Trans. Signal Process..

[16]  D. Fuhrmann.  An algorithm for subspace computation, with applications in signal processing , 1988 .

[17]  S. T. Smith.  Intrinsic Cramer-Rao bounds and subspace estimation accuracy , 2000, Proceedings of the 2000 IEEE Sensor Array and Multichannel Signal Processing Workshop. SAM 2000 (Cat. No.00EX410).

[18]  Anuj Srivastava,et al.  A Bayesian approach to geometric subspace estimation , 2000, IEEE Trans. Signal Process..

[19]  Jun S. Liu,et al.  Sequential Monte Carlo methods for dynamic systems , 1997 .

[20]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[21]  Joos Vandewalle,et al.  An improved stochastic gradient algorithm for principal component analysis and subspace tracking , 1997, IEEE Trans. Signal Process..