Minimum mean square distance estimation of a subspace

TL;DR

This paper introduces a Bayesian subspace estimation method that minimizes the mean square distance on the Grassmann manifold, providing more natural and accurate estimates than traditional MSE-based approaches, especially with limited data.

Contribution

It proposes a novel MMSD estimator for subspace estimation on the Grassmann manifold, derived from the posterior distribution, and demonstrates its effectiveness through examples and an application to hyperspectral imagery.

Findings

MMSD estimator outperforms traditional methods with fewer samples

The estimator is derived analytically or via MCMC in various models

Accurate subspace estimates achieved in hyperspectral imagery application

Abstract

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 $\hat{U}$ 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 $U U^{T}$. 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.