A Bayesian approach to geometric subspace estimation

This paper presents a geometric approach to estimating subspaces as elements of the 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(n)]. Subspace estimation or tracking then corresponds to inferences on U(n). Taking a Bayesian approach, a posterior density is derived on U(n), and certain expectations under this posterior are empirically generated. For the choice of the Hilbert-Schmidt norm on U(n), 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.