A Bayesian approach to geometric subspace estimation

This paper presents a geometric approach to estimating subspaces as elements of complex Grassmann-manifold, with each subspace represented by its unique, complex projection matrix. Variation between the subspaces is modeled by rotating their projection matrices via the action of unitary matrices [elements of the unitary group U( )]. Subspace estimation or tracking then corresponds to inferences on U( ). Taking a Bayesian approach, a posterior density is derived on U( ), and certain expectations under this posterior are empirically generated. For the choice of the Hilbert–Schmidt norm on U( ), to define estimation errors, an optimal MMSE estimator is derived. It is shown that this estimator achieves a lower bound on the expected squared errors associated with all possible estimators. The estimator and the bound are computed using (Metropolis-adjusted) Langevin’s-diffusion algorithm for sampling from the posterior. For use in subspace tracking, a prior model on subspace rotation, that utilizes Newtonian dynamics, is suggested.