Robust dimension reduction, fusion frames, and Grassmannian packings

Abstract

We consider estimating a random vector from its noisy projections onto low-dimensional subspaces constituting a fusion frame. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first determine the minimum mean-squared error (MSE) in linearly estimating the random vector of interest from its fusion frame projections, in the presence of white noise. We show that MSE assumes its minimum value when the fusion frame is tight. We then analyze the robustness of the constructed linear minimum MSE (LMMSE) estimator to erasures of the fusion frame subspaces. We prove that tight fusion frames consisting of equi-dimensional subspaces have maximum 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 subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures. We call such fusion frames equi-distance tight fusion frames, and prove that the chordal distance between subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for construction of equi-distance tight fusion frames.

Gabor fusion frames generated by difference sets

“…Tightness is an important property, required for example, for minimization of the recovery error of a random vector from its noisy fusion frame measurements [19]. Among other desirable properties are equidimensionality and equidistance.…”

Constructing fusion frames with desired parameters

“…2,11,20,22 It also provides robustness to subspace perturbations, 12 which may arise due to imprecise knowledge of sensor network topology. In most cases, extra structure on fusion frames is required to guarantee satisfactory performance.…”

“…In most cases, extra structure on fusion frames is required to guarantee satisfactory performance. For instance, our recent work 20,22 shows that in order to minimize the mean-squared error in the linear minimum mean-squared error estimation of a random vector from its fusion frame measurements in white noise the fusion frame needs to be Parseval or tight. The Parseval property is also desirable for managing signal processing complexity.…”

“…Other examples of optimality of structured fusion frames for signal reconstruction can be found in. 2,3,11,12,20,22 A natural question is: how can one construct fusion frames with desired fusion frame operators? More specifically, how can one construct fusion frames for which the fusion frame operator has a desired set of eigenvalues?…”

“…In frame theory, * a signal is represented by a collection of scalars, which measure the amplitudes of the projections of the signal onto the frame vectors, whereas in fusion frame theory the signal is represented by the projections of the signal onto the fusion frame subspaces. In a two-stage data processing setup, these projections serve as locally processed data, which can be combined to reconstruct a signal of interest (see, e.g., 13,20,23 ).…”