A geometric approach to subspace tracking

A geometric approach to the subspace tracking problem is introduced. We propose to use a dynamic model for subspace motion in the subspace estimation problem, and thereby make possible the estimation of signal subspaces over data collection intervals for which the subspace cannot be assumed constant. The development of the dynamic model requires the introduction of Grassmann manifolds or their equivalent, the spaces of N/spl times/N complex projection matrices of rank M. The geometry of these spaces, including representations of the tangent space and geodesic trajectories, is described. The problem of estimating a geodesic trajectory from a data sequence is set forth. The maximum-likelihood solution for the M=1 case is derived, under the assumption that the 2-dimensional subspace containing the geodesic is given.