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Bayesian estimation of a subspace

We consider the problem of subspace estimation in a Bayesian setting. First, we revisit the conventional minimum mean square error (MSE) estimator and explain why the MSE criterion may not be fully suitable when operating in the Grassmann manifold. As an alternative, we propose to carry out subspace estimation by minimizing the mean square distance between the true subspace U and its estimate, where the considered distance is a natural metric on the Grassmann manifold. We show that the resulting estimator is no longer the posterior mean of U but entails computing the principal eigenvectors of the posterior mean of UUT. Illustrative examples involving a linear Gaussian model for the data and a Bingham or von Mises Fisher prior distribution for U are presented. In the former case the minimum mean square distance (MMSD) estimator is obtained in closed-form while, in the latter case, a Markov chain Monte Carlo method is used to approximate the MMSD estimator. The method is shown to provide accurate estimates even when the number of samples is lower than the dimension of U. Finally, an application to hyperspectral imagery is presented.