Fusion frames and robust dimension reduction

Abstract

We consider the linear minimum mean- squared error (LMMSE) estimation of a random vector of interest from its fusion frame measurements in presence noise and subspace erasures. Each fusion frame measurement is a low-dimensional vector whose elements are inner products of an orthogonal basis for a fusion frame subspace and the random vector of interest. We derive bounds on the mean-squared error (MSE) and show that the MSE will achieve its lower bound if the fusion frame is tight. We prove that tight fusion frames consisting of equi- dimensional subspaces have maximum robustness with respect to erasures of one subspace, and that the optimal dimension depends on SNR. We also show that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures, and refer to such fusion frames as equi-distance tight fusion frames. Finally, we show that the squared chordal distance between the subspaces in such fusion frames meets the so-called simplex bound, and thereby establish a connection between equidistance tight fusion frames and optimal Grassmannian packings.