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Minimum mean square distance estimation of a subspace

Besson, Olivier and Dobigeon, Nicolas and Tourneret, Jean-Yves Minimum mean square distance estimation of a subspace. (2011) IEEE Transactions on Signal Processing, 59 (12). 5709 -5720. ISSN 1053-587X

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Official URL: http://dx.doi.org/10.1109/TSP.2011.2166548


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 U may not be adequate as the MSE is not the natural metric in the Grassmann manifold G.N.p, i.e., the set of p-dimensional subspaces in RN. As an alternative, we propose to carry out subspace estimation by minimizing the mean square distance 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 UUT. Derivation of the minimum mean square distance (MMSD) estimator is carried out in a few illustrative examples including a linear Gaussian model for the data and Bingham or von Mises Fisher prior distributions 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.

Item Type:Article
Additional Information:Thanks to IEEE editor. (c) 2011 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. The definitive version is available at http://ieeexplore.ieee.org
HAL Id:hal-03610507
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE)
Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UT3 (FRANCE)
Université de Toulouse > Université Toulouse - Jean Jaurès - UT2J (FRANCE)
Université de Toulouse > Université Toulouse 1 Capitole - UT1 (FRANCE)
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Deposited On:25 Nov 2011 15:23

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