Robust dimension reduction, fusion frames, and Grassmannian packings
Robust dimension reduction, fusion frames, and Grassmannian packings
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DOI：
10.1016/j.acha.2008.03.001
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发表时间：
2007-09
影响因子：
2.5
通讯作者：
Gitta Kutyniok;A. Pezeshki;Robert Calderbank;Taotao Liu
中科院分区：
文献类型：
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作者：
Gitta Kutyniok;A. Pezeshki;Robert Calderbank;Taotao Liu
我们考虑在存在噪声和子空间擦除的融合帧中从其测量值估计随机向量。融合框架是子空间的集合，子空间上的投影算子的和上下由恒等算子的常数倍有界。我们首先考虑在存在加性白噪声的情况下，从随机感兴趣向量的融合帧测量中得到线性最小均方误差（LMMSE）估计。每个融合帧测量是一个向量，其元素是融合帧子空间的正交基与感兴趣的随机向量的内积。我们推导了均方误差（MSE）的边界，并证明了当融合框架较紧时，均方误差将达到其下界。然后分析了所构造的LMMSE估计量对融合框架子空间擦除的鲁棒性。我们将擦除分析限制在紧融合框架的范围内，并假设所有擦除都同等重要。在这些假设下，我们证明了由等维子空间组成的紧密融合框架对于所有紧密融合框架中一个子空间的擦除具有最大的鲁棒性（在MSE意义上），并且最优子空间维数取决于信噪比（SNR）。我们还证明了由具有相等对弦距离的等维子空间组成的紧密融合框架对于两个或两个以上的子空间擦除是最鲁棒的。我们称这种融合框架为等距紧密融合框架。我们证明了这种融合框架中子空间的弦距平方满足所谓的单纯形界，从而建立了等距离紧密融合框架与最优格拉斯曼填充之间的联系。最后，我们给出了几个构造等距离紧密融合框架的例子。
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 mean-squared 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 mean-squared 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 equi-dimensional 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 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, among the class of equi-dimensional tight fusion frames. We call such fusion frames equi-distance tight fusion frames. We prove that the squared chordal distance between the 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 the construction of equi-distance tight fusion frames.