Fusion frames and robust dimension reduction
We consider the linear minimum meansquared 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 equidimensional 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. © 2008 IEEE.
