[Banach] Abstract of a paper by Gitta Kutyniok, Ali Pezeshki, Robert Calderbank, and Taotao Liu

[Dale Alspach]

This is an announcement for the paper "Robust dimension reduction,
fusion frames, and Grassmannian packings" by Gitta Kutyniok, Ali
Pezeshki, Robert Calderbank, and Taotao Liu.


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.

Archive classification: math.FA

Mathematics Subject Classification: 94A12; 42C15; 68P30; 93E10

Remarks: 21 pages

The source file(s), fusionframe_final_arxiv.bbl: 2844 bytes

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 http://front.math.ucdavis.edu/0709.2340

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