Math @ Duke

Publications [#235908] of Robert Calderbank
Papers Published
 Kutyniok, G; Pezeshki, A; Calderbank, R; Liu, T, Robust dimension reduction, fusion frames, and Grassmannian packings,
Applied and Computational Harmonic Analysis, vol. 26 no. 1
(2009),
pp. 6476, ISSN 10635203 [doi]
(last updated on 2018/10/21)
Abstract: We consider estimating a random vector from its measurements in a fusion frame, in presence of noise and subspace erasures. 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 consider the linear minimum meansquared error (LMMSE) estimation of the random vector of interest from its fusion frame measurements in the presence of additive white noise. Each fusion frame measurement is a 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 meansquared error (MSE) and show that the MSE will achieve its lower bound if the fusion frame is tight. We then analyze the robustness of the constructed LMMSE estimator to erasures of the fusion frame subspaces. We limit our erasure analysis to the class of tight fusion frames and assume that all erasures are equally important. Under these assumptions, we prove that tight fusion frames consisting of equidimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace among all tight fusion frames, and that the optimal subspace dimension depends on signaltonoise ratio (SNR). We also prove that tight fusion frames consisting of equidimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures, among the class of equidimensional tight fusion frames. We call such fusion frames equidistance tight fusion frames. We prove that the squared chordal distance between the subspaces in such fusion frames meets the socalled simplex bound, and thereby establish connections between equidistance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for the construction of equidistance tight fusion frames. © 2008 Elsevier Inc. All rights reserved.


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