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# Minimum mean square distance estimation of a subspace

### Bingham/von Mises Fisher prior-based estimator

### CS decomposition-based estimator

### Application to hyperspectral imagery

We consider the problem of subspace estimation in a Bayesian setting. Since we are operating in the Grassmann manifold, the usual approach which consists of minimizing the mean square error (MSE) between the true subspace U and its estimate may not be adequate as the MSE is not the natural metric in the Grassmann manifold. As an alternative, we propose to carry out subspace estimation by minimizing the mean square distance (MSD) between U and its estimate, where the considered distance is a natural metric in the Grassmann manifold, viz. the distance between the projection matrices. 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 U U^{T}.

First, derivation of the MMSD estimator is carried out in a few illustrative examples including a linear Gaussian model for the data and a Bingham or von Mises Fisher prior distribution for U. In all scenarios, posterior distributions are derived and the MMSD estimator is obtained either analytically or implemented via a Markov chain Monte Carlo simulation method. The method is shown to provide accurate estimates even when the number of samples is lower than the dimension of U. An application to hyperspectral imagery is finally investigated.

The Bayesian subspace estimation algorithm and the main results have been published in IEEE Trans. Signal Processing:

- article .

An extended version of this paper (with additional proofs and algorithms) is available as a technical report:

The main idea consists of using the CS decomposition of the semi-orthogonal matrix H whose columns span the subspace of interest. This parametrization is intuitively appealing and allows for non informative prior distributions of the matrices involved in the CS decomposition and very mild assumptions about the angles between the actual subspace and the prior subspace. The posterior distributions are derived and a Gibbs sampling scheme is presented to obtain the minimum mean-square distance estimator of the subspace of interest. Numerical simulations and an application to real hyperspectral data assess the validity and the performances of the estimator.

The CS decomposition based estimator is described in the following paper published in IEEE Trans. Signal Processing:

- article .

Additional results are available in the related technical report:

- mpg file [ .mpg - 2124Ko ].