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We consider estimating a random vector from its noisy projections onto lowdimensional subspaces constituting a fusion frame. A fusion frame is acollection of subspaces, for which the sum of the projection operators onto thesubspaces is bounded below and above by constant multiples of the identityoperator. We first determine the minimum mean-squared error (MSE) in linearlyestimating the random vector of interest from its fusion frame projections, inthe presence of white noise. We show that MSE assumes its minimum value whenthe fusion frame is tight. We then analyze the robustness of the constructedlinear minimum MSE (LMMSE) estimator to erasures of the fusion frame subspaces.We prove that tight fusion frames consisting of equi-dimensional subspaces havemaximum robustness (in the MSE sense) with respect to erasures of one subspace,and that the optimal subspace dimension depends on signal-to-noise ratio (SNR).We also prove that tight fusion frames consisting of equi-dimensional subspaceswith equal pairwise chordal distances are most robust with respect to two andmore subspace erasures. We call such fusion frames equi-distance tight fusionframes, and prove that the chordal distance between subspaces in such fusionframes meets the so-called simplex bound, and thereby establish connectionsbetween equi-distance tight fusion frames and optimal Grassmannian packings.Finally, we present several examples for construction of equi-distance tightfusion frames.

Robust Dimension Reduction, Fusion Frames, and Grassmannian Packings