Robust Dimension Reduction, Fusion Frames, and Grassmannian Packings

TL;DR

This paper analyzes optimal fusion frames for estimating signals from noisy projections, demonstrating that tight, equi-dimensional, and equi-distance frames minimize MSE and maximize robustness, linking them to Grassmannian packings.

Contribution

It characterizes the optimal properties of tight fusion frames with equi-dimensional and equi-distance subspaces for robust signal estimation, establishing new connections to Grassmannian packings.

Findings

Tight fusion frames minimize MSE in noisy estimation.

Equi-distance tight fusion frames maximize robustness to subspace erasures.

Chordal distances in optimal frames meet the simplex bound.

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.