鲁棒降维、融合框架和Grassmannian填充
Robust Dimension Reduction, Fusion Frames, and Grassmannian Packings
摘要：
我们考虑估计一个随机向量从其噪声投影到低维子空间构成一个融合帧。融合框架是子空间的集合，对于该集合，子空间上的投影算子的和由恒等算子的常数倍数上下限定。我们首先确定在存在白噪声的情况下从其融合帧投影线性估计感兴趣的随机向量的最小均方误差（MSE）。我们表明，MSE假设其最小值时，融合框架是紧的。然后，我们分析了所构造的线性最小均方误差（LMMSE）估计器的融合帧子空间的擦除的鲁棒性。我们证明了紧密的融合框架组成的等维子空间具有最大的鲁棒性（在MSE意义上）相对于一个子空间的擦除，和最佳的子空间维数取决于信噪比（SNR）。我们还证明了紧密的融合框架组成的等维子空间具有相等的成对弦距离是最强大的两个和更多的子空间擦除。我们称这种融合框架为等距紧融合框架，并证明了这种融合框架中子空间的弦距离满足单纯形界，从而建立了等距紧融合框架与最优Grassmannian填充之间的联系.最后给出了几个等距紧融合框架的构造实例。
英文摘要：
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.
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